<s>[INST] 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 $P(x^2-1)$.
 [/INST] Here is another way. Let $t :=\sqrt{x-1}$. Then note that 
$$\begin{align}
t&=\sqrt{x-1};\\
t^2&=x-1;\\
t^2+1&=x;\\
t^4&=x^2-2x+1.
\end{align}$$
So that $$\begin{align}
P(x)=P(t^2+1)&=t^4+5t^2+3\\
&=(x^2-2x+1)+5(x-1)+3\\
&=x^2+3x-1.
\end{align}.$$
Lastly, evaluate $P(x^2-1)=\ldots=x^4+x^2-3$.
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