A function takes in an input and returns an output.
In this function g(x)g(x), you input xx and it returns x^3-2x^2x3−2x2. We can also look at this as this: you input xx (what's enclosed by the parentheses) and this function g(x)g(x) takes the input (x), cubes it (x^3x3), subtracts two times the input squared (-2x^2−2x2). Putting this all together, this particular function g(x)g(x) takes in an input xx and returns the expression x^3-2x^2x3−2x2, with xx representing the input (which could be anything).
Cool! So what exactly does the question mean by g(x+1)g(x+1)? What this means is that the entire expression x+1x+1 is the input! So, because
g(x)=x^3-2x^2g(x)=x3−2x2
so,
g(x+1)=(x+1)^3-2(x+1)^2g(x+1)=(x+1)3−2(x+1)2
So now, all we have to do is use algebra to simplify this expression! We have
g(x+1)=(x+1)^3-2(x+1)^2g(x+1)=(x+1)3−2(x+1)2
g(x+1)=(x+1)(x+1)(x+1)-2(x^2+2x+1)g(x+1)=(x+1)(x+1)(x+1)−2(x2+2x+1)
g(x+1)=(x+1)(x^2+2x+1)-2(x^2+2x+1)g(x+1)=(x+1)(x2+2x+1)−2(x2+2x+1)
Here, we see that we have a (x^2+2x+1)(x2+2x+1) term in common on both sides of the minus sign. We invoke the distributive property:
g(x+1)=(x^2+2x+1)(x+1-2)g(x+1)=(x2+2x+1)(x+1−2)
g(x+1)=(x^2+2x+1)(x-1)g(x+1)=(x2+2x+1)(x−1)
And we're done!
SIDENOTE:
Alternatively, you could also write the answer like this:
g(x+1)=(x+1)^2(x-1)=(x^2-1)(x+1)g(x+1)=(x+1)2(x−1)=(x2−1)(x+1)
Either way is fine.
