$\begin{bmatrix} a\\\\b\end{bmatrix} \longmapsto \begin{bmatrix}0&1\\\\-4&-4\end{bmatrix}\begin{bmatrix} a\\\\b\end{bmatrix}$
Use the characteristic polynomail to find all eigenvalues for the transformation for each eigenvalues $\lambda$ find all eigenvectors with eigenvalues $\lambda$ and find a basis for $E_\lambda$
Here is what I have so far
$\operatorname{char}(f)\begin{bmatrix}-x&1\\\\-4&-4-x\end{bmatrix}=x^2+4x+4 = (x+2)^2$ so there is one eigenvalue $\lambda=-2$
I'm not sure how to find a basis for $E_{-2}$ though so this is as far as I have got.