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The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. In algebra, it involves finding the largest expression that is common in the terms of a polynomial.
For example, in the expression given:
\[m(m + 2n) + n(m + 2n)\]
we look for the common factors in each term.
Here, both terms have a common expression, which is (m + 2n).
Recognizing this common factor allows us to simplify or factor the entire expression more easily.
Finding the GCF simplifies algebraic expressions and makes calculations easier. It's a crucial step in the factoring process.

Factored form is the representation of an expression as a product of its factors. It helps us understand the structure of algebraic expressions.
In our case, we have already determined the common factor: (m + 2n).
By factoring out (m + 2n) from each term in the expression:
\[m(m + 2n) + n(m + 2n) = (m + 2n)(m + n)\]
We notice that we rewrite the original expression in a simpler, factored form.
Factoring expressions involves several steps:

• Identifying the common factors.
• Rewriting the expression as a product.
• Simplifying the factored form, if possible.

Being comfortable with factoring is important for solving algebra problems efficiently.

Algebraic expressions are combinations of variables, numbers, and operations (addition, subtraction, multiplication, division). In our example:
\[m(m + 2n) + n(m + 2n)\]
we see an algebraic expression composed of variables (m, n), numbers, and operations.
Recognizing patterns and common factors in algebraic expressions is key to simplifying them.
For instance, noting that (m + 2n) appears in both terms helps us simplify the expression:
\[m(m + 2n) + n(m + 2n) = (m + 2n)(m + n)\]
This form shows how the original complex expression can be broken down into simpler parts.
Developing a strong understanding of algebraic expressions enables us to tackle more complex problems in math and apply these skills in real-life scenarios.