Q:

What is $f(x)-f(x + 2)$ if $f(x + 1) = x(x + 2)$?

What is $f(x)-f(x + 2)$ if $f(x + 1) = x(x + 2)$?

I have tired but I just cant seem to get my head around it.

A:

Here's a hint.
You know that $f(x+1) = x(x+2).$
Let's figure out $f(x)$. Since $x$ is one less than $x+1$, every place we see $x$ on the right side, we replace it with $x-1$:
$f((x+1)-1) = f(x) = (x-1)[(x-1)+2] = (x-1)(x+1) = x^2-1.$
To find $f(x+2)$, simply replace with $(x+1)$ instead.
Can you take it from here?
Here's another way of looking at it.
Let's rewrite the equation with $y$ instead of $x$, since it doesn't matter what we call the variable:
$$f(y+1) = y(y+2).$$
All I did above in the first part was substitute $y=x-1$. This gives me $f(x)$ because $f(y+1) = f((x-1)+1) = f(x)$.
To get the other part you need, which is $f(x+2)$, you simply let $y = x+1$ and do the same thing.

