The equation x² + px + q = 0 has real p and q.

Step 1. The pair. The roots are not real, since two real roots at distance $5$ from $0$ with positive real parts would be equal. So they are $u \pm vi$ with $v > 0$.

Step 2. The gap. They are $2v$ apart, so $2v = 6$ and $v = 3$.

Step 3. The distance from 0. $u^2 + 9 = 25$, so $u = 4$, taking the positive real part.

Step 4. Sum and product. The sum is $8 = -p$, so $p = -8$. The product is $|4 + 3i|^2 = 25$, so $q = 25$.