Is it Polynomial division?

The roots of $x^2 + 2x + 5$ are $-1 \pm 2i$ , $i$ being the imaginary unit.

So, $-1 \pm 2i$ must be roots of $x^4 + Px^2 + Q$ too, which leads to:

$(-1 \pm 2i)^4 + P(-1 \pm 2i)^2 + Q = 0$

$-7 \pm 24i + P(-3 ∓ 4i) + Q = 0$

Then we have:

$24 - 4P = 0$

$Q = 7 + 3P$

Which result in:

$P = 6, Q = 25, P+Q=\fbox{31}$