The problem requires filling in the blanks to factor linear expressions. We need to find the missing terms in the following expressions: $5x + 30 = 5(x + \_\_)$ $4cd + 18c = \_\_c(2d + 9)$ $12 + 20w = 4(\_\_ + 5w)$

1. Problem Description

The problem requires filling in the blanks to factor linear expressions. We need to find the missing terms in the following expressions:

$5x + 30 = 5(x + \_\_)$

$4cd + 18c = \_\_c(2d + 9)$

$12 + 20w = 4(\_\_ + 5w)$

2. Solution Steps

For the first expression, $5x + 30 = 5(x + \_\_)$, we can factor out a 5 from both terms:

$5x + 30 = 5(x + 6)$ since $5*6 = 30$. So the missing term is

6.

For the second expression, $4cd + 18c = \_\_c(2d + 9)$, we can factor out $c$ from both terms:

$4cd + 18c = c(4d + 18)$. Now we have $c(4d + 18) = \_\_c(2d + 9)$.

We notice that $2*(2d + 9) = 4d + 18$, so we can deduce that the missing term is

2. Thus, $4cd + 18c = 2c(2d + 9)$.

For the third expression, $12 + 20w = 4(\_\_ + 5w)$, we can factor out a 4 from both terms:

$12 + 20w = 4(3 + 5w)$ since $4*3 = 12$ and $4*5w = 20w$. So the missing term is

3.

3. Final Answer

$5x + 30 = 5(x + 6)$

$4cd + 18c = 2c(2d + 9)$

$12 + 20w = 4(3 + 5w)$