Question

1. rewrite each of the following expressions as the product of two binomials by factoring out a common binomial factor. watch out for the subtraction problems (b) and (d).

(a) ((x + 5)(x + 1) + (x + 5)(x + 8))
(b) ((2x - 1)(3x + 5) - (2x - 1)(x + 4))
(c) ((x - 7)(x - 9) + (x - 7)(4x + 5))
(d) ((x + 1)(5x - 7) - (x + 1)(x - 3))

Explanation:

(a) $(x + 5)(x + 1) + (x + 5)(x + 8)$

We can factor out the common binomial factor $(x + 5)$ from both terms.

Step 1: Identify the common binomial factor

The common binomial factor in both terms is $(x + 5)$.

Step 2: Factor out the common binomial factor

Using the distributive property $ab + ac = a(b + c)$, we have:

$$
LATEXBLOCK0
$$

Step 3: Simplify the expression inside the brackets

$$
LATEXBLOCK1
$$

We can factor out the common binomial factor $(2x - 1)$ from both terms.

Step 1: Identify the common binomial factor

The common binomial factor in both terms is $(2x - 1)$.

Step 2: Factor out the common binomial factor

Using the distributive property $ab - ac = a(b - c)$, we have:

$$
LATEXBLOCK0
$$

Step 3: Simplify the expression inside the brackets

$$
LATEXBLOCK1
$$

We can factor out the common binomial factor $(x - 7)$ from both terms.

Step 1: Identify the common binomial factor

The common binomial factor in both terms is $(x - 7)$.

Step 2: Factor out the common binomial factor

Using the distributive property $ab + ac = a(b + c)$, we have:

$$
LATEXBLOCK0
$$

Step 3: Simplify the expression inside the brackets

$$
LATEXBLOCK1
$$

Answer:

$(x + 5)(2x + 9)$

(b) $(2x - 1)(3x + 5) - (2x - 1)(x + 4)$