For all values of x, f(x)=(x+1)^2 and g(x)=2(x-1). Show that gf(x)=2x(x+2)

The question is asking us to find gf(x).
When approaching this question, the first thing I tend to look for is what information we have that we can break down. In this case, I can immediately see that the f(x) is a quadratic equation (ie an equation with an x^2 value, an x value and an integer).
Since we are looking for gf(x), we are going to have to substitute f(x) into the g(x) formula. This substitution an important part of what we are doing.
This is where our understanding of a quadratic equation is now useful. We take the g(x)=2(x-1) equation and use the f(x) formula to substitute values of x. g(x) becomes g(x)=2((x+1)^2-1). Be careful with use of brackets here.
We now work through this algebra. It might be best to expand the (x+1)^2 first. This becomes (x+1)(x+1)=x^2+2x+1. We can now substitute this back into g(x), which is now g(x)=2(x^2+2x+1)-1)=2(x^2+2x). We are now very close to the expression we are looking for! All we have to do is move the x outside of the bracket - simplifying the equation. g(x)=2x(x+2). We have now shown that gf(x)=2x(x+2).