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When factoring by grouping, one of the first steps is to identify the greatest common factor (GCF) within pairs of terms. The GCF is the largest factor that divides two or more numbers or terms. For algebraic expressions, the GCF can be a variable, a number, or both. Determining the GCF helps to simplify the expression, making it easier to factor.
For example, in the expression given in the exercise,
First group: \(cd + 6c\): The GCF is \(c\) because both terms contain \(c\).
Second group: \(4d + 24\): The GCF is \(4\) because both terms are divisible by \(4\).
Finding the GCF is essential as it helps in breaking down more complex expressions, leading to easier and more understandable factorizations.

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. These can range from simple (e.g., \(x + 2\)) to more complex expressions (e.g., \(3x^2 - 4xy + 7\)). They are the building blocks in algebra, used in equations and functions.
In the exercise, the expression we are working with is \(cd + 6c + 4d + 24\). This expression involves variables \(c\) and \(d\) and coefficients \(6\), \(4\), and \(24\).
Understanding how to manipulate and simplify these expressions is key to solving algebra problems. Manipulation often includes techniques like combining like terms and applying the distributive property, both of which are necessary for factoring.

Binomial factorization is the process of breaking down a polynomial into two-term (binomial) factors. This concept becomes handy when the expression in question can be seen as the product of simpler binomials. Factoring by grouping often results in binomial factors.
In the provided exercise, after factoring out the GCF from each grouped pair, we analyze: \[(cd + 6c) + (4d + 24)\] to obtain \[c(d + 6) + 4(d + 6)\].
Here, \((d + 6)\) serves as the common binomial factor. Factoring it out gives the result: \[(c + 4)(d + 6)\].
Recognizing and factoring out binomial expressions simplifies the polynomial, making it more manageable and solving the problem easier.