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The greatest common factor (GCF) is the largest factor that two or more numbers or terms have in common.
This concept is crucial when simplifying algebraic expressions by factoring. In the given exercise, we identify the common factor in both parts of the expression.
Here’s how we determine it step-by-step:
The expression is: \( q(q + 4p) + p(q + 4p) \)
Observe the terms carefully. Both terms have a common algebraic factor of \( (q + 4p) \).
By factoring out the GCF, in this case, \( (q + 4p) \), we simplify the expression. Remember, identifying the GCF not only simplifies calculations but also reveals the structure of algebraic expressions more clearly, making them easier to work with.

Factored form of an algebraic expression refers to expressing it as a product of its factors.
Factoring makes complex expressions simpler and reveals their underlying structure. For the given problem, we start with:
\( q(q + 4p) + p(q + 4p) \)
First, we identify the common term, which is \( (q + 4p) \)
Next, we factor out \( (q + 4p) \), giving us:
\( (q + 4p)(q + p) \)
So, the factored form of the expression is \( (q + 4p)(q + p) \).
This shows the expression as a product of two factors. By rewriting expressions in factored form, algebraic operations like solving equations, simplifying expressions, and finding zeroes become more straightforward.

Algebraic expressions are combinations of variables, numbers, and arithmetic operations.
They form the foundation of algebra and are used to represent real-life situations in mathematical terms. In the given exercise, understanding algebraic expressions is crucial:
The starting expression is:
\( q(q + 4p) + p(q + 4p) \)
This is a combination of variables \( q \) and \( p \) and arithmetic operations like addition and multiplication. The process of factoring helps in simplifying complex algebraic expressions. By rewriting the expression in a factored form, we simplify it to:
\( (q + 4p)(q + p) \)
This makes it easier to interpret and solve. Always remember to look for common patterns, like terms that can be factored out, to simplify algebraic expressions efficiently.