Let $L$ be the linear transformation from the vector space of polynomials of degree $2$ to the vector space of $22$ matrices satisfying $L(1-x+x^2)=\left[\begin{array}{cc}a& a\\ -b& 2\end{array}\right],L(-1+x+3x^2)=\left[\begin{array}{cc}b& -2\\ c& 2\end{array}\right]$ and $L(x^2)=\left[\begin{array}{cc}0& 1\\ 1& d\end{array}\right].$ Find $a,b,c$ and $d$