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Factoring polynomials can be challenging, but the grouping method is a helpful technique to simplify the process.
When using the grouping method, you rearrange and divide the polynomial into groups of terms.
In the given exercise, the polynomial is divided into two pairs:

• Group 1: \(x^2 + 3x\)
• Group 2: \(2x + 6\)

The reason for grouping is to help you identify patterns or common factors within smaller portions of the polynomial.
In our case, each group is easier to manage and can be individually factored by extracting the greatest common factor (GCF).
This method can be particularly useful when dealing with four-term polynomials, as it allows you to break them down into solvable chunks.

The greatest common factor (GCF) is an important concept when factoring polynomials, as it allows you to simplify expressions.
It represents the largest number or variable term that can evenly divide every term in a polynomial.
Identifying the GCF is crucial in the factorization process as it makes the expression simpler and more manageable.
In the context of the exercise, consider each group individually:

• For the first group \( (x^2 + 3x) \), the factor both terms share is \( x \). Factor \( x \) out to transform the expression into \( x(x + 3) \).
• For the second group \( (2x + 6) \), the terms share the factor \( 2 \). Extract \( 2 \) to get \( 2(x + 3) \).

By factoring the GCF, you create a simpler expression that brings you closer to your final solution.

The common binomial factor emerges once each grouped part of the polynomial has been reduced.
It leverages the repetition of a binomial within the factored groups.
In this exercise, both groups after factoring shared a common binomial factor \((x + 3)\).
Finding and factoring out this common binomial is the final step in simplifying the polynomial expression.
Once you've identified the common binomial factor, you can factor it out, similar to how you would with a single term.
In the exercise, once the binomial \((x + 3)\) is factored out, the expression simplifies to:

• \((x + 3)(x + 2)\)

This gives us a neatly packaged product of factors, indicating the solution to the original polynomial expression.