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 $P(x^2-1)$.

One more alternative is to start by deducing that $P(x)$ is a quadratic based on the fact that its degree has to be $2$ in order to output a quartic when the argument is a quadratic expression. So let $P(x) = ax^2 + bx + c$.
When $x^2 + 1 = 0, x = \pm i$, so we get From $P(0) = 1 - 5 + 3 = -1$, and also $P(0) = c$ we can immediately get $c=-1$.
When $x=0, x^2 + 1  = 1$ so we have that $P(1) = 3$ and $P(1) = a+b-1$, so $a+b = 4$.
When $x = 1, x^2 + 1 = 2$, so we have that $P(2) = 9$ and $P(2) = 4a + 2b - 1$, so $4a + 2b = 10$.
Solving the latter two for $a$ and $b$, we get $a=1, b=3$.
Hence $P(x) = x^2 + 3x - 1$ and $P(x^2-1)$ can be worked out.
