I have a linear transformation
given by 
And I need to find its spectral decomposition.
I want to represent T as a matrix so I could find it's eigenvalues , and then calculate their eigenspaces or use Lagrange's interpolation.
I'm not sure if I should represent in base 
or in some other base. And if it is in that base, I don't understand how i should look like because of the parameters a,b,c and the given form, because I only saw more simple examples
like this:
So I'll be glad to see how is the representative matrix of T looks like and how do I see from it the eigenvalues ?
Look at $T$ with respect to the basis $\{ x^2, x, 1\}$. Say, $$ \left[\begin{array}{c} a \\ b \\ c\end{array}\right] \sim ax^2+bx+c. $$ Then $T(ax^2+bx+c)=4ax^2+2bx+2a+3b$ translates to the following matrix formulation $$ \left[ T \right]\left[\begin{array}{c}a \\ b \\ c\end{array}\right] = \left[\begin{array}{c}4a\\2b\\2a+3b\end{array}\right]=\left[\begin{array}{ccc}4 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 3 & 0\end{array}\right]\left[\begin{array}{c}a \\ b \\ c\end{array}\right] $$ The characteristic polynomial of $T$ is $$ \det\left[\begin{array}{ccc}\lambda-4 & 0 & 0 \\ 0 & \lambda-2 & 0 \\ -2 & -3 & \lambda\end{array}\right] = (\lambda-4)(\lambda-2)\lambda $$ There is a basis of eigenvectors $\{ 1,2x+3,2x^2+1 \}$ with corresponding eigenvalues $\{0,2,4\}$. That is, $$ (T-0I)(1) =0 \\ (T-2I)(2x+3)=0 \\ (T-4I)(2x^2+1)=0. $$