$T:\mathbb{R}_2[x] \mapsto \mathbb{R}_2[x]$ s.t.:
$$ \begin{array}{l} T(1) = 3+2x+4x^2, \\ T(x) = 2+2x^2, \\ T(x^2) = 4+2x+3x^2. \end{array} $$
Is there base $B$ of $\mathbb{R}_2[x]$ that $[T]_B = \begin{pmatrix} 2 & 0 & 0 \\-1 & 1 & 0 \\ 5 & 0 & 4 \end{pmatrix}$?