You have already a fine answer. If completing the square is not easy for you, there is an algorithm for exactly this purpose. In the matrix answer repeated just below, note that your quadratic form is doubled ($H$ is the Hessian matrix) and writing it out gives something very similar to the answer you accepted.
$$\frac{y_1^2-y_2^2-y_3^2+y_4^2}4$$
$$y_1=x_1+x_2+x_3+2x_4,$$
$$y_2=-x_1+x_2+x_3,$$
$$y_3=x_3+2x_4,$$
$$y_4=x_3.$$
$$ Q^T D Q = H $$
$$\left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & - 1 & 0 & 0 \\
\frac{ 1 }{ 2 } & 1 & 0 & 0 \\
\frac{ 1 }{ 2 } & 1 & \frac{ 1 }{ 2 } & 1 \\
1 & 0 & 1 & 0 \\
\end{array}
\right)
\left(
\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & - \frac{ 1 }{ 2 } & 0 & 0 \\
0 & 0 & - 2 & 0 \\
0 & 0 & 0 & \frac{ 1 }{ 2 } \\
\end{array}
\right)
\left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 1 \\
- 1 & 1 & 1 & 0 \\
0 & 0 & \frac{ 1 }{ 2 } & 1 \\
0 & 0 & 1 & 0 \\
\end{array}
\right)
= \left(
\begin{array}{rrrr}
0 & 1 & 1 & 1 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
\end{array}
\right)
$$
Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia
$$ H = \left(
\begin{array}{rrrr}
0 & 1 & 1 & 1 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
\end{array}
\right)
$$
$$ D_0 = H $$
$$ E_j^T D_{j-1} E_j = D_j $$
$$ P_{j-1} E_j = P_j $$
$$ E_j^{-1} Q_{j-1} = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$
$$ H = \left(
\begin{array}{rrrr}
0 & 1 & 1 & 1 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
\end{array}
\right)
$$
==============================================
$$ E_{1} = \left(
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{1} = \left(
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; Q_{1} = \left(
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
- 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; D_{1} = \left(
\begin{array}{rrrr}
2 & 1 & 1 & 2 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 \\
\end{array}
\right)
$$
==============================================
$$ E_{2} = \left(
\begin{array}{rrrr}
1 & - \frac{ 1 }{ 2 } & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{2} = \left(
\begin{array}{rrrr}
1 & - \frac{ 1 }{ 2 } & 0 & 0 \\
1 & \frac{ 1 }{ 2 } & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; Q_{2} = \left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 0 & 0 \\
- 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; D_{2} = \left(
\begin{array}{rrrr}
2 & 0 & 1 & 2 \\
0 & - \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & 0 \\
1 & - \frac{ 1 }{ 2 } & 0 & 0 \\
2 & 0 & 0 & 0 \\
\end{array}
\right)
$$
==============================================
$$ E_{3} = \left(
\begin{array}{rrrr}
1 & 0 & - \frac{ 1 }{ 2 } & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{3} = \left(
\begin{array}{rrrr}
1 & - \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & 0 \\
1 & \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; Q_{3} = \left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 0 \\
- 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; D_{3} = \left(
\begin{array}{rrrr}
2 & 0 & 0 & 2 \\
0 & - \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & 0 \\
0 & - \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & - 1 \\
2 & 0 & - 1 & 0 \\
\end{array}
\right)
$$
==============================================
$$ E_{4} = \left(
\begin{array}{rrrr}
1 & 0 & 0 & - 1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{4} = \left(
\begin{array}{rrrr}
1 & - \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & - 1 \\
1 & \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & - 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; Q_{4} = \left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 1 \\
- 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; D_{4} = \left(
\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & - \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & 0 \\
0 & - \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & - 1 \\
0 & 0 & - 1 & - 2 \\
\end{array}
\right)
$$
==============================================
$$ E_{5} = \left(
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & - 1 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{5} = \left(
\begin{array}{rrrr}
1 & - \frac{ 1 }{ 2 } & 0 & - 1 \\
1 & \frac{ 1 }{ 2 } & - 1 & - 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; Q_{5} = \left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 1 \\
- 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; D_{5} = \left(
\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & - \frac{ 1 }{ 2 } & 0 & 0 \\
0 & 0 & 0 & - 1 \\
0 & 0 & - 1 & - 2 \\
\end{array}
\right)
$$
==============================================
$$ E_{6} = \left(
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{array}
\right)
$$
$$ P_{6} = \left(
\begin{array}{rrrr}
1 & - \frac{ 1 }{ 2 } & - 1 & 0 \\
1 & \frac{ 1 }{ 2 } & - 1 & - 1 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{array}
\right)
, \; \; \; Q_{6} = \left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 1 \\
- 1 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{array}
\right)
, \; \; \; D_{6} = \left(
\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & - \frac{ 1 }{ 2 } & 0 & 0 \\
0 & 0 & - 2 & - 1 \\
0 & 0 & - 1 & 0 \\
\end{array}
\right)
$$
==============================================
$$ E_{7} = \left(
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & - \frac{ 1 }{ 2 } \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{7} = \left(
\begin{array}{rrrr}
1 & - \frac{ 1 }{ 2 } & - 1 & \frac{ 1 }{ 2 } \\
1 & \frac{ 1 }{ 2 } & - 1 & - \frac{ 1 }{ 2 } \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & - \frac{ 1 }{ 2 } \\
\end{array}
\right)
, \; \; \; Q_{7} = \left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 1 \\
- 1 & 1 & 1 & 0 \\
0 & 0 & \frac{ 1 }{ 2 } & 1 \\
0 & 0 & 1 & 0 \\
\end{array}
\right)
, \; \; \; D_{7} = \left(
\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & - \frac{ 1 }{ 2 } & 0 & 0 \\
0 & 0 & - 2 & 0 \\
0 & 0 & 0 & \frac{ 1 }{ 2 } \\
\end{array}
\right)
$$
==============================================
$$ P^T H P = D $$
$$\left(
\begin{array}{rrrr}
1 & 1 & 0 & 0 \\
- \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 0 & 0 \\
- 1 & - 1 & 0 & 1 \\
\frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } & 1 & - \frac{ 1 }{ 2 } \\
\end{array}
\right)
\left(
\begin{array}{rrrr}
0 & 1 & 1 & 1 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
\end{array}
\right)
\left(
\begin{array}{rrrr}
1 & - \frac{ 1 }{ 2 } & - 1 & \frac{ 1 }{ 2 } \\
1 & \frac{ 1 }{ 2 } & - 1 & - \frac{ 1 }{ 2 } \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & - \frac{ 1 }{ 2 } \\
\end{array}
\right)
= \left(
\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & - \frac{ 1 }{ 2 } & 0 & 0 \\
0 & 0 & - 2 & 0 \\
0 & 0 & 0 & \frac{ 1 }{ 2 } \\
\end{array}
\right)
$$
$$ Q^T D Q = H $$
$$\left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & - 1 & 0 & 0 \\
\frac{ 1 }{ 2 } & 1 & 0 & 0 \\
\frac{ 1 }{ 2 } & 1 & \frac{ 1 }{ 2 } & 1 \\
1 & 0 & 1 & 0 \\
\end{array}
\right)
\left(
\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & - \frac{ 1 }{ 2 } & 0 & 0 \\
0 & 0 & - 2 & 0 \\
0 & 0 & 0 & \frac{ 1 }{ 2 } \\
\end{array}
\right)
\left(
\begin{array}{rrrr}
\frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } & 1 \\
- 1 & 1 & 1 & 0 \\
0 & 0 & \frac{ 1 }{ 2 } & 1 \\
0 & 0 & 1 & 0 \\
\end{array}
\right)
= \left(
\begin{array}{rrrr}
0 & 1 & 1 & 1 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
\end{array}
\right)
$$