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To understand factoring by grouping, it's essential to know what a greatest common factor (GCF) is. The GCF of a set of terms is the largest factor that divides each of the terms without leaving a remainder.

In the context of the exercise, we first group the terms into pairs:

• First pair: 16q^2 + 20q
• Second pair: -28q - 35

For the first pair, the GCF is 4q, because 4q is the largest term that can divide both 16q^2 and 20q.

For the second group, the GCF is -7, because -7 is the largest term that divides both -28q and -35.

Factoring out the GCF from each pair of terms helps simplify the expression and makes it easier to find common binomial factors in the next steps.

A binomial factor consists of two terms. In the context of factoring by grouping, binomial factors are key to solving the problems.

After factoring out the GCF from each pair, the exercise's expression turns into:

• 4q(4q + 5) - 7(4q + 5)

Here, (4q + 5) is the common binomial factor in both grouped terms.

Factoring by grouping leverages these common binomial factors to further simplify the expression. Once identified, you can factor them out.

In this case, you factor out the common binomial factor (4q + 5) from each term, leading to: (4q + 5)(4q - 7)

Grouping terms is the first step in the factoring by grouping method. By strategically grouping terms that share common factors, we can simplify the process of factoring complex expressions.

Let's revisit the exercise: 16q^2 + 20q - 28q - 35
First, group the terms into pairs:

• (16q^2 + 20q) - (28q + 35)

This step is vital because it sets up the expression for easier factorization. The goal is to have each group have a GCF that can be factored out.

After grouping, the individual pairs can be simplified by factoring out the GCF from each pair. This simplification leads to the identification of a common binomial factor, which can then be factored out, making the entire expression manageable.

Understanding how to group terms effectively is crucial for mastering factoring by grouping.