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When working with algebraic expressions, identifying the common factor is a crucial first step in the factoring process. A common factor is a term or expression that is shared by all components of a polynomial or algebraic expression.
This shared factor can be a constant, a variable, or even a combination of terms.

• In the given exercise, the expression is: \[ 2x(y+4) + 3(y+4) \]. • Notice how both terms in the expression include the factor \((y+4)\). Recognizing a common factor helps simplify expressions, making them easier to rewrite as a product of two binomials. Once identified, the common factor can be pulled out, leaving behind a simpler expression to work with. Understanding this concept is fundamental when dealing with more complex algebraic simplifications.

A binomial is an expression that contains two terms, typically separated by a plus or minus sign.
When we talk about the product of binomials, we are referring to a result from multiplying two binomial expressions together.

For example, in the step-by-step solution from the original exercise, we took the expression:\[(y+4)(2x + 3)\]This is a representation of multiplying two binomials:• The first binomial is \((y+4)\).• The second binomial is \((2x+3)\).The process of representing complex expressions as the product of binomials allows for simplification and is a foundational step in solving algebraic equations.
By breaking down expressions into simpler parts, you'll find it easier to work with and solve equations.

Factoring expressions involves breaking down a complex expression into simpler factors that, when multiplied, result in the original expression.
This process can simplify the handling of expressions, especially when solving equations or simplifying fractions.

In the exercise we are looking into, the expression:\[2x(y+4) + 3(y+4)\]was simplified by factoring out the common term \[(y+4)\],leaving us with:\[(y+4)(2x + 3)\].The key steps in factoring expressions include:• Identifying any common factors present.• Rewriting the expression in terms of these factors.• Rechecking your work to ensure accuracy.Understanding how to factor expressions is a primary skill in algebra that aids in simplifying tasks ranging from arithmetic simplifications to calculus problems.