Exercise 56 Page 533

We want to completely factor the given expression. To do so, we will first identify and factor out the greatest common factor.

Factor Out the GCF

The greatest common factor (GCF) of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. The GCF of the given expression is $2.$

Factor the Quadratic Trinomial

Here we have a quadratic trinomial of the form $a{x}^2+bx+c,$ where $|a|\ne 1$ and there are no common factors. To factor this expression, we will rewrite the middle term, $bx,$ as two terms. The coefficients of these two terms will be factors of $ac$ whose sum must be $b.$ We have that $a=6,$ $b=11,$ and $c=\text{-}7.$ There are now three steps we need to follow in order to rewrite the above expression.

1. Find $\mathbf{a}\mathbf{c}.$ Since we have that $a=6$ and $c=\text{-}7,$ the value of $ac$ is $6\times \text{-}7=\text{-}42.$
2. Find factors of $\mathbf{a}\mathbf{c}.$ Since $ac=\text{-}42,$ which is negative, we need factors of $ac$ to have opposite signs — one positive and one negative — in order for the product to be negative. Since $b=11,$ which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor, so that their sum is positive.

3. Rewrite $\mathbf{b}\mathbf{x}$ as two terms. Now that we know which factors are the ones to be used, we can rewrite $bx$ as two terms.
   $$\begin{array}{c}2(6x^2+11x-7)\\ \Updownarrow \\ 2(6x^2-3x+14x-7)\end{array}$$

Finally, we will factor the last expression obtained.