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The distributive property is a very helpful algebraic rule. It lets us multiply each term inside a set of parentheses by a term outside the parentheses. This property is essential when dealing with expressions involving binomials. For instance, with the expression \((y^n + 4)(y^n - 5)\), you apply the distributive property by multiplying each term in the first binomial by each term in the second binomial. As detailed in the solution: \(y^n \times y^n + y^n \times (-5) + 4 \times y^n + 4 \times (-5)\). Breaking it down, you're distributing each term and ensuring every combination is covered.

After using the distributive property, you'll often have multiple similar terms to deal with. Like terms are terms that have the same variable raised to the same power. For example, in our expression \(y^n \times (-5)\text{ and }4 \times y^n\), both are like terms because they contain \(y^n\). In the solution: \( y^{2n} - 5y^n + 4y^n - 20 \), you notice two like terms: \(-5y^n\text{ and }4y^n\). Combining like terms simplifies the expression further, making math much more manageable. So, when we combine \(-5y^n\text{ and }4y^n\), we get \(- y^n\), simplifying the expression to \( y^{2n} - y^n - 20 \).

Binomial multiplication involves multiplying two binomials. Binomials are algebraic expressions with two terms. Our given example is \((y^n + 4)(y^n - 5)\). Here, each term from the first binomial multiplies every term in the second binomial. This procedure is often remembered by the acronym FOIL - First, Outside, Inside, Last. This stands for multiplying the first terms of each binomial, outside terms, inside terms, and the last terms. This helps ensure that you’ve covered all multiplications.
For our example, following the FOIL method sequences exactly how we handle each pair of terms. The resulting terms from FOIL are \(y^n \times y^n\), \(y^n \times (-5)\), \(4 \times y^n\), and finally \(4 \times (-5)\).