How has this been simplified?

Question

I have tried to undersatnd how this step works, it shows a rather messy equation on the top line. Then the line below is one step taken to simplfy. But I can't quite figure out what has happend.

If anyone could shed some light on this it would be extremely helpful.

Answer 1

In the above expression, you have a confusing expression:

$$36x\left(9x^2+3\right)\left(2x-8\right)-2\left(9x^2+3\right)^2$$

One way to see what happened is to simplify it by letting $y=9x^2+3$, and let $z=2x-8$ then substitute to get:

$$36x(y)(z)-2(y^{2})$$

We did the above step to be able to "see" the factors clearly.

Now, factor out $(2y)$ to get:

$$(2y)(18xz-y)$$

At this point you could substitute back the $y$ and $z$ to get the result.

Answer 2

It's just a matter of arithmetic: $$\frac{36x(9x^2+3)(2x-8)-2(9x^2+3)^2}{(2x-8)^2}=\frac{ 18x\cdot \color{#66F}{2(9x^2+3)}(2x-8)-\color{#66F} {2(9x^2+3)}(9x^2+3)}{(2x-8)^2}= $$ $$=\frac{ 2(9x^2+3)[18x(2x-8)- (9x^2+3)]}{(2x-8)^2} $$