(v^2+1)^2-(v^2-2v-1)^2
Rule for squaring an expression with more than two terms.
Viz. Get the square of every term and two times all possible products.
e.g. (a+b+c+d)*2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd
So now square the two expressioms in the question. This gives us,
v^4 + 1 + 2v^2 - (v^4 + 4v^2 + 1 - 4v^3 - 2v^2 + 4v)
2v^2 - 4v^2 + 4v^3 + 2v^2 - 4v
4v^3 - 4v
4v(v^2 - 1)
4v(v + 1)(v - 1)