Factor each polynomial completely. See Example 6. 6ar + 12br - 5as - 10bs

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Group the terms in pairs to make factoring easier: $(6ar + 12br) - (5as + 10bs)$.

Factor out the greatest common factor (GCF) from each group separately. From the first group $(6ar + 12br)$, factor out \$6r$, and from the second group $(5as + 10bs)$, factor out $5s$.

After factoring out the GCFs, rewrite the expression as \$6r(a + 2b) - 5s(a + 2b)$.

Notice that both terms now contain the common binomial factor $(a + 2b)$. Factor this binomial out.

Write the completely factored form as $(a + 2b)(6r - 5s)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring by Grouping

Factoring by grouping involves rearranging and grouping terms in a polynomial to find common factors within each group. This method helps simplify the expression by factoring out the greatest common factor (GCF) from each group, making it easier to factor the entire polynomial.

Greatest Common Factor (GCF)

The GCF is the largest factor that divides two or more terms in a polynomial. Identifying the GCF in each group of terms is essential for factoring by grouping, as it allows you to simplify terms and reveal common binomial factors.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It is used in reverse during factoring to extract common factors from terms. Recognizing how to apply this property helps in rewriting polynomials as products of factors.

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