What is the complete factorisation of \(x^4+2x^2+1\)?

Correct answer: \((x^2+1)^2\)

The direct answer is A: \(x^4+2x^2+1=(x^2+1)^2\). Treat \(x^2\) as one algebraic quantity, say \(X=x^2\). The expression becomes \(X^2+2X+1\), which has the identity \(X^2+2X+1=(X+1)^2\). Replacing \(X\) by \(x^2\) gives \((x^2+1)^2\). Option A is therefore correct. B would expand to \(x^4+4x^3+6x^2+4x+1\), containing terms that are not present. C expands to \(x^4-2x^2+1\), so its middle sign is wrong. D is equal to \((x^2-1)^2\), and therefore also has the negative middle term \(-2x^2\). The key idea is to notice the first and last terms as squares: \(x^4=(x^2)^2\) and \(1=1^2\), then check that the middle term is twice their product, namely \(2x^2\).