Find bases for the kernel and range of the linear transformation $S$ below. \begin{equation*} \ltdefn{S}{M_{22}}{P_2},\quad\lt{S}{\begin{bmatrix}a&b\\c&d\end{bmatrix}}= (a+2b+5c-4d)+(3a-b+8c+2d)x+(a+b+4c-2d)x^2 \end{equation*}