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Binomials are expressions in algebra that consist of two terms. These terms are typically separated by a plus or minus sign. For example, in the exercise \((x + 5y)(7x + 3y)\), each group within parentheses is a binomial. Understanding binomials is crucial because they form the basic building blocks for more complex algebraic expressions.
Here’s a quick breakdown:

• The first binomial is \(x + 5y\), consisting of two terms: \(x\) and \(5y\).
• The second binomial is \(7x + 3y\), with the terms \(7x\) and \(3y\).

Recognizing binomials within algebraic expressions allows you to apply various algebraic methods to simplify or expand them. The FOIL method, which we will use in this problem, is specifically designed for multiplying binomials together.

Polynomial multiplication might sound complicated, but the FOIL Method makes it easy when dealing with two binomials. FOIL stands for First, Outer, Inner, and Last, which describes the pairs of terms you multiply together.
In our exercise, with binomials \((x + 5y)\) and \((7x + 3y)\), here’s a step-by-step application of FOIL:

• First: Multiply the first terms in each binomial: \(x \cdot 7x = 7x^2\)
• Outer: Multiply the outer terms of the binomials: \(x \cdot 3y = 3xy\)
• Inner: Multiply the inner terms: \(5y \cdot 7x = 35xy\)
• Last: Multiply the last terms of each binomial: \(5y \cdot 3y = 15y^2\)

After applying the FOIL method, we collected the products: \(7x^2\), \(3xy\), \(35xy\), and \(15y^2\). These products form the expanded expression \(7x^2 + 3xy + 35xy + 15y^2\).

Once you've expanded a polynomial or multiplied binomials, the next step is to simplify the expression by combining like terms. Like terms are terms that have identical variable parts raised to the same power.
In the expression from our exercise, \(7x^2 + 3xy + 35xy + 15y^2\), the terms \(3xy\) and \(35xy\) are like terms because they both have the variable part \(xy\).
Here's how to combine them:

• Add the coefficients of these like terms together: \(3xy + 35xy = 38xy\).
• Now, the expression simplifies to \(7x^2 + 38xy + 15y^2\).

By combining like terms, you’re reducing the expression to its simplest form, which is often necessary for solving algebraic equations and for clearly understanding the results of polynomial operations. This process is fundamental in algebra and helps in making the problem more manageable.