Question

How do you factor `x^2 + 7x + 8x + 56` by grouping?

Answer 1

`x=-8` & `x=-7`

Explanation:

Let's look at the `x^2+7x` and the `8x+56` separately.

`color(red)(x^2+7x)+color(blue)(8x+56)=0`

We can factor an `x` out of the red term, and an `8` out of the blue term. We get:

`color(red)(x(x+7))+color(blue)(8(x+7))=0`

Since `x+7` is multiplying two terms, we can rewrite this as:

`(x+8)(x+7)=0`

Setting the factors equal to zero, we get:

`x=-8` & `x=-7`

Answer 2

`(x+8)(x+7)`

Explanation:

This expression is already in the perfect form for grouping. Note that `x^2+7x = x(x+7)` and `8x+56 = 8(x+7)` so that both pairs of terms in `color(red)(x^2+7x)+color(blue)(8x+56)` have a common factor :

`color(red)(x^2+7x)+color(blue)(8x+56) = color(red)(x(x+7))+color(blue)(8(x+7))`

You can now use the law `bcolor (red)a+c color(red)a = (b+c)color(red)a` to write this as

`(x+8)(x+7)`