Suppose that $a$ and $b$ are nonzero real numbers, and that the equation  $${x^2 + ax + b = 0}$$ has solutions $a$ and $b$. Then what is the pair $(a,b)$?
The given conditions imply that $$
x^2 + ax + b = (x-a)(x-b) = x^2 -(a+b)x + ab,
$$ so $$
a+b = -a \quad\text{and}\quad ab = b.
$$ Since $b \neq 0$, the second equation implies that $a=1$. The first equation gives $b=-2$, so $(a,b) = \boxed{(1,-2)}$.