Find all points on the XY plane where parabola doesn't pass

Find all points on the XY plane where parabola $y=x^2-4px+2p^2-3$ doesn't pass. Note that $P\in \mathbb{R}$.

Answer below

$y<-x^2$

Deep explanation would be awesome because I can't do anything in this problem.

Solution 1:

$$f(x)=x^2-4px+2p^2-3$$Suppose $(a,b)$ is a point the parabola cannot pass through for any value of $p$. We want to find the locus of this point. Consider the equation $$f(a)=b \\ a^2-4pa+2p^2-3=b \\ 2p^2-4pa+a^2-b-3=0$$ The condition is that this quadratic in $p$ should have no solution, i.e. for any value of $b$ there should be no value of $a$ such that this equation holds for any $p$. $$ D\lt 0 \\ \implies16a^2-8(a^2-b-3)\lt 0 \\ \implies b\lt -a^2-3$$ I believe the answer you gave has a typo.