Let me make a tiny-microscopic improvement. Let $n := n(x)$. Then
$$ P(x)\ |\ (n-1)\cdot(n-2)\ =\ (n-\frac 32)^2 - \frac 14$$
It follows that:
THEOREM
$$ n(x)\quad \ge\quad \left\lceil\sqrt{P(x)+\frac 14}\ +\ \frac 32\right\rceil $$
Let me make a tiny-microscopic improvement. Let $n := n(x)$. Then
$$ P(x)\ |\ (n-1)\cdot(n-2)\ =\ (n-\frac 32)^2 - \frac 14$$
It follows that:
THEOREM
$$ n(x)\quad \ge\quad \left\lceil\sqrt{P(x)+\frac 14}\ +\ \frac 32\right\rceil $$