Show that a determinant equals the product of another determinant and a polynomial function without calculating Show without calculating the determinant, that 
$$
    \det\left(\begin{bmatrix}
    a_{1}+b_{1}x & a_{1}-b_{1}x & c_{1}\\
    a_{2}+b_{2}x & a_{2}-b_{2}x & c_{2}\\
    a_{3}+b_{3}x & a_{3}-b_{3}x& c_{3}\\
    \end{bmatrix}\right) 
    = 
    -2x 
    \det\left(\begin{bmatrix}
    a_{1} & b_{1}x & c_{1}\\
    a_{2} & b_{2}x & c_{2}\\
    a_{3} & b_{3}x& c_{3}\\
    \end{bmatrix}\right)
$$ 
I've tried using the multilinearity of the determinant, but i didn't get very far and now im stuck. 
 A: $
\left|\begin{matrix}
a_1+b_1x & a_1-b_1x & c_1\\
a_2+b_2x & a_2-b_2x & c_2\\
a_3+b_3x & a_3-b_3x & c_3
\end{matrix}\right|$
$=\;\; \left|\begin{matrix}
a_1 & a_1-b_1x & c_1\\
a_2 & a_2-b_2x & c_2\\
a_3 & a_3-b_3x & c_3
\end{matrix}\right|+\left|\begin{matrix}
b_1x & a_1-b_1x & c_1\\
b_2x & a_2-b_2x & c_2\\
b_3x & a_3-b_3x & c_3
\end{matrix}\right|$
$=\;\; \left|\begin{matrix}
a_1 & a_1 & c_1\\
a_2 & a_2 & c_2\\
a_3 & a_3 & c_3
\end{matrix}\right|+\left|\begin{matrix}
a_1 & -b_1x & c_1\\
a_2 & -b_2x & c_2\\
a_3 & -b_3x & c_3
\end{matrix}\right|+\left|\begin{matrix}
b_1x & a_1 & c_1\\
b_2x & a_2 & c_2\\
b_3x & a_3 & c_3
\end{matrix}\right|+\left|\begin{matrix}
b_1x & -b_1x & c_1\\
b_2x & -b_2x & c_2\\
b_3x & -b_3x & c_3
\end{matrix}\right|
$
$=\;\; 0+\left|\begin{matrix}
a_1 & -b_1x & c_1\\
a_2 & -b_2x & c_2\\
a_3 & -b_3x & c_3
\end{matrix}\right|+\left|\begin{matrix}
b_1x & a_1 & c_1\\
b_2x & a_2 & c_2\\
b_3x & a_3 & c_3
\end{matrix}\right|+0
$
$=\;\; -x \left|\begin{matrix}
a_1 & b_1 & c_1\\
a_2 & b_2 & c_2\\
a_3 & b_3 & c_3
\end{matrix}\right|+ x\left|\begin{matrix}
b_1 & a_1 & c_1\\
b_2 & a_2 & c_2\\
b_3 & a_3 & c_3
\end{matrix}\right|
$
$=\;\; -x \left|\begin{matrix}
a_1 & b_1 & c_1\\
a_2 & b_2 & c_2\\
a_3 & b_3 & c_3
\end{matrix}\right|-x \left|\begin{matrix}
a_1 & b_1 & c_1\\
a_2 & b_2 & c_2\\
a_3 & b_3 & c_3
\end{matrix}\right|
$
$=\;\; -2x \left|\begin{matrix}
a_1 & b_1 & c_1\\
a_2 & b_2 & c_2\\
a_3 & b_3 & c_3
\end{matrix}\right|
$
A: It is indeed multilinearity you want to use. Write
$$
a =
\begin{bmatrix}
a_{1}\\
a_{2}\\
a_{3}
\end{bmatrix},
\quad
b =
\begin{bmatrix}
b_{1}\\
b_{2}\\
b_{3}
\end{bmatrix},
\quad
c =
\begin{bmatrix}
c_{1}\\
c_{2}\\
c_{3}
\end{bmatrix},
$$
then
\begin{align*}
\det(\begin{bmatrix}a + b x & a - b x & c\end{bmatrix})
&=
\det(\begin{bmatrix}a & a & c\end{bmatrix})
- x^{2}
\det(\begin{bmatrix}b & b & c\end{bmatrix})
\\&- x
\det(\begin{bmatrix}a & b & c\end{bmatrix})
+ x
\det(\begin{bmatrix}b & a & c\end{bmatrix}).
\end{align*}
