jmc

Seeing that factoring will not work, we apply the Quadratic Formula: $$\begin{array}{rl}x& =\frac{-(5)\pm \sqrt{(5{)}^2-4(1)(8)}}{2(1)}\\ & =\frac{-5\pm \sqrt{25-32}}{2}=\frac{-5\pm \sqrt{-7}}{2}=-\frac{5}{2}\pm \frac{\sqrt{7}}{2}i.\end{array}$$ Now we see that $a=-\frac{5}{2}$ and $b=\pm \frac{\sqrt{7}}{2},$ so $a+b^2=-\frac{5}{2}+\frac{7}{4}=\boxed{-\frac{3}{4}}.$