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When you encounter problems involving the multiplication of two binomials, the FOIL method is a technique you'll frequently use. FOIL is an acronym that stands for First, Outer, Inner, Last, which represents the order in which you multiply the terms.

The process involves four steps:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the innermost terms.
• Last: Multiply the last terms in the binomials.

Once the multiplications are complete, what follows is a combination of like terms to simplify the expression fully. This step is crucial as it leads to a simplified version of the algebraic expression that's easier to understand and work with in future calculations.

Combining like terms is the process of simplifying an algebraic expression by merging terms that have the same variables raised to the same powers. For instance, in the expression \(4x^2 - 10x + 6x - 15\), both \( -10x \) and \( +6x \) are like terms since they both contain the variable \( x \) raised to the first power.

To combine them, you add the coefficients (the numerical parts) of these like terms while keeping the variable part unchanged. This step condenses the expression, making it less cluttered and more straightforward for subsequent operations or for understanding the underlying relationships between the variables and constants in the expression.

Multiplication of binomials involves expanding expressions that consist of two terms. A binomial is an algebraic expression with two unlike terms, for example, \(2x + 3\). When you multiply binomials, you are essentially using the distributive property to expand the product into a polynomial.

This expansion usually results in a quadratic expression, which may look daunting at first. However, by systematically applying the FOIL method, the complex process breaks down into simpler steps that are easy to execute. If we use binomial multiplication on our original exercise, \(2x + 3\)(\(2x - 5\)), we evenly distribute each term in the first binomial across the second binomial. This leads to four terms, which when simplified by combining like terms, results in a manageable quadratic expression that captures the product of the original binomials.