Let a and b be such that $-2A^2 + 4A -3I_2 = aA + bI_2$. To find $a + b$. $A \in M_2$ with characteristic polynomial $p(x) = x^2 -3x - 5$. Let a and b be such that  $-2A^2 + 4A -3I_2 = aA + bI_2$. To find $a + b$.

A matrix satisfies its characteristic polynomial.
$$-2A^2+4A-3I=aA+bI$$
$$-2A^2+(4-a)A+(-3-b)I=0$$
$$-2\left(A^2+\frac{a-4}{2}A+\frac{3+b}{2}I\right)=0$$
$$A^2+\frac{a-4}{2}A+\frac{3+b}{2}I=0$$
From this we need:
$$\frac{a-4}{2}=-3$$
$$a-4=-6$$
$$a=-2$$
and
$$\frac{3+b}{2}=-5$$
$$3+b=-10$$
$$b=-13$$
So $a+b=-15$
