Question

How do you multiply and simplify `\frac { x ^ { 2} + 5x - 14} { x ^ { 2} + 6x - 16} \cdot \frac { 2x + 16} { x + 6}`?

Answer 1

`(2x+14)/(x+6)`

Explanation:

Factor numerators and denominators to get

`((x+7)(x-2))/((x+8)(x-2)) xx (2(x+8))/(x+6).`

Cancel like factors to get"

`((x +7) xx2)/(x+6)`

`= (2x+14)/(x+6)`

Answer 2

`(2x+14)/(x+6)`

Explanation:

First, let's factor everything that can be factored. I'm going to look at each set of terms individually. Note that `x+6` cannot be factored.

`x^2+5x-14=(x+7)(x-2)`

`x^2+6x-16=(x+8)(x-2)`

`2x+16=2(x+8)`

Now that we have the factored terms, let's put these factors into the expression.

`((x+7)(x-2))/((x+8)(x-2))*(2(x+8))/(x+6)`

Now that we have the factored terms, we see that the numerator and denominator have `x-2` and `x+8` in common. These terms will cancel out.

`((x+7)(x-2))/((x+8)(x-2))*(2(x+8))/(x+6) ->`

`((x+7)(cancel(x-2)))/((cancel(x+8))(cancel(x-2)))*(2(cancel(x+8)))/(x+6) ->`

When you cancel the terms out, you get the following.

`(2(x+7))/(x+6)=(2x+14)/(x+6)`