In which form can (x^2-y^2+2x+1) be factorised?

Correct answer: (x+1-y)(x+1+y)

The correct answer is A: \((x+1-y)(x+1+y)\). Rearrange the terms so that \(x^2+2x+1\) is together. This is the perfect square \((x+1)^2\), so the expression becomes \((x+1)^2-y^2\). Applying \(X^2-Y^2=(X-Y)(X+Y)\), with \(X=x+1\) and \(Y=y\), gives \((x+1-y)(x+1+y)\). Option A expands to \((x+1)^2-y^2\) and hence reproduces \(x^2+2x+1-y^2\). Option B expands to \(x^2-y^2+1\) and misses the term \(2x\). Option C is the square of \(x+y+1\) and would contain positive \(2xy\) and \(2y\) terms, which are absent. Option D represents \((x-1)^2-y^2\) and would give \(-2x\) rather than \(+2x\). Memory cue: the sign of the middle term tells whether the square is \((x+1)^2\) or \((x-1)^2\).