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When factoring algebraic expressions, it's fundamental to first identify the greatest common factor (GCF). The GCF is the largest factor that divides all terms in a group without leaving a remainder. This helps simplify the equations and make them more manageable.

In the example given, we saw two groups:

• \((6y^2 + 7y)\)
• \((24y + 28)\)

For each group, you need to find the GCF. For the first group \((6y^2 + 7y)\), the GCF is \(y\), because it is common in both terms. For the second group \((24y + 28)\), the GCF is \(4\).

This process simplifies the problem, as seen in our solution:

• \(6y^2 + 7y = y(6y + 7)\)
• \(24y + 28 = 4(6y + 7)\)

A binomial is an algebraic expression containing exactly two terms. The term 'binomial' comes from 'bi', meaning 'two', and 'nomial', which refers to a term or component. In simpler words, binomials are expressions like \((a + b)\) or \((x - y)\).

In the example exercise, we recognize the binomial \((6y + 7)\). After factoring the GCF from each group, this binomial appears in both terms of the expression. The concept of binomials is significant because it allows us to notice common factors and to further factorize the entire expression.

Thus, we see in the solution that after factoring out the GCF, the reduced expression becomes:

• \(y(6y + 7) + 4(6y + 7)\)

Here, \((6y + 7)\) is our common binomial, enabling the next step of factorization.

Understanding algebraic expressions is crucial for mastering topics like factoring by grouping. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation signs but does not have an equality sign.

In the given problem, \(6y^2 + 7y + 24y + 28\) is an algebraic expression consisting of terms linked by addition. To simplify or solve it, different strategies like factoring by grouping are used.

When factoring by grouping, you first isolate groups of terms and simplify them, which was done here by splitting the expression into

• \((6y^2 + 7y)\)
• \((24y + 28)\)

Then, you factor out the GCF from each group, recognizing binomials to factor further:

• \(y(6y + 7) + 4(6y + 7)\)

This step breaks down complex algebraic expressions into simpler, factorable components ensuring easier problem-solving processes.