Let T:P3→P3 be the linear transformation such that $T(−2x^2)= −2x^2 − 2x$, $T(0.5x + 2)= 3x^2 + 4x−2$, and $T(2x^2 − 1)= 2x + 1$. Find $T(1), T(x), T(x^2)$, and $T(ax^2 + bx + c)$, where a, b, and c are arbitrary real numbers.
I understand how to find $T(x^2)$ where you just divide the given $T(-2x^2)$ by $-2$ to get $T(x^2) = x^2 + x$.
I'm not sure sure how to proceed in calculating the other transformation functions. Please list as many of the steps as possible in solving for the three other functions.