Exercise 142 Page 515

In this case, we have $\text{-}15.$ This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite sign — one positive and one negative.

Factor Constants	Product of Constants
$1$ and $\text{-}15$	$\text{-}15$
$\text{-}1$ and $15$	$\text{-}15$
$3$ and $\text{-}5$	$\text{-}15$
$\text{-}3$ and $5$	$\text{-}15$

Next, let's consider the coefficient of the linear term. For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, $\text{-}2.$

Factors	Sum of Factors
$1$ and $\text{-}15$	$\text{-}14$
$\text{-}1$ and $15$	$14$
$3$ and $\text{-}5$	$\text{-}2$
$\text{-}3$ and $5$	$2$

We found the factors whose product is $\text{-}15$ and whose sum is $\text{-}2.$

Factor Out The GCF

The GCF of an expression is the common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. In this case, the GCF is $7.$ The result of factoring out a GCF from the given expression is a binomial with a leading coefficient of $1.$ Let's temporarily only focus on this binomial, and we will bring back the GCF after factoring.

Factor the Expression

Do you notice that the binomial is a $\text{difference}$ of two $\text{perfect}$ $\text{squares}?$ This can be factored using the difference of squares method. To do so, we first need to express each term as a perfect square.

Expression	$x^2-9$
Rewrite as Perfect Squares	$x^2-3^2$
Apply the Formula	$(x+3)(x-3)$

Wait! Before we finish, remember that we factored out a GCF from the original expression. To fully complete the factored expression, let's reintroduce that GCF now.

We have that $a=3,$ $b=10,$ and $c=8.$ 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 $a=3$ and $c=8,$ the value of $ac$ is $3\times 8=24.$
2. Find factors of $\mathbf{a}\mathbf{c}.$ Since $ac=24,$ which is positive, we need factors of $ac$ to have the same sign — both positive or both negative — in order for the product to be positive. Since $b=10,$ which is also positive, those factors will need to be positive 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.
   $$\begin{array}{c}3x^2+10x+8\\ \Updownarrow \\ 3x^2+4x+6x+8\end{array}$$

Finally, we will factor the last expression obtained.