Let $T:P_3 \rightarrow P_3$ be the linear transformation such that $$T(-2 x^2)= 2 x^2 + 3 x, \ \ \ T(0.5 x + 5)= 3 x^2 + 4 x - 3, \ \ \ T(4 x^2 - 1) = -3 x + 3 .$$ Find $T(1)$, $T(x)$, $T(x^2)$, and $T(a x^2 + b x + c)$, where $a$, $b$, and $c$ are arbitrary real numbers.

$T(1)=$ ,
$T(x)=$ ,
$T(x^2)=$ ,
$T(a x^2 + b x + c)=$ .