Given that $P(x)$ is a polynomial such that $P(x^2+1) = x^4+5x^2+3$, what is $P(x^2-1)$? How would I go about solving this? I can't find a clear relation between $x^2+1$ and $x^4+5x^2+3$ to solve $P(x^2-1)$.

Hint:   let $y=x^2$, then the given condition writes as $P(y+1)=y^2+5y+3$ and it follows that $P(x^2-1)=P\big((y-2)+1\big)= (y-2)^2+5(y-2)+3 = \ldots$
