Question 165
Determine all pairs of real numbers $a$ and $b$, $b \gt 0$, such that the solutions to the two equations
$$
x^2 + a x + a = b
$$
and
$$
x^2 + a x + a = -b
$$
are four consecutive integers.
$$
x^2 + a x + a = b
$$
and
$$
x^2 + a x + a = -b
$$
are four consecutive integers.