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Binomials are expressions that have exactly two terms. These terms are usually connected by a plus (+) or minus (-) sign. In the given exercise, we have two binomial expressions:

• y + 3
• y + 4

Binomials are a fundamental part of algebra and are often used in solving quadratic equations, factoring, and of course, polynomial expansions. Recognizing and working with binomials is essential because it allows us to use various algebraic methods like the distributive property to simplify and solve complex problems. To identify a binomial, look for two distinct terms, separated by either a plus or minus sign, enclosed in parentheses.

The Distributive Property is a key concept in algebra. It is used to expand expressions using multiplication over addition or subtraction. Sometimes expressed through FOIL (First, Outer, Inner, Last) in the context of binomials, it tells us how to multiply two binomial expressions together.In the case of our problem, we apply the distributive property as follows:

• First, multiply the first term of each binomial: \(y \times y = y^2\).
• Next, multiply the outer terms: \(y \times 4 = 4y\).
• Then, multiply the inner terms: \(3 \times y = 3y\).
• Finally, multiply the last terms: \(3 \times 4 = 12\).

The distributive property allows us to multiply two polynomials efficiently, simplifying what might otherwise be a cumbersome calculation.It helps break down the expansion into manageable arithmetic steps.

After using the distributive property, we often end up with terms that need further simplification. Combining like terms is the process of merging terms that have the same variable raised to the same power. This is important because it simplifies the expression and helps us to see the overall structure more clearly.In the expanded expression, \(y^2 + 4y + 3y + 12\), we notice that there are two terms that can be combined:

• \(4y\) and \(3y\).

Adding these together, we get \(7y\). The expression then simplifies to: \(y^2 + 7y + 12\).
Simplifying by combining like terms is often the last step when expanding polynomials. It not only cleans up the expression, making it easier to read, but also provides the simplest form of the expression, which is crucial for solving or further manipulating algebraic expressions.