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The distributive property is a fundamental rule in algebra that allows us to multiply a single term by each term within a parenthesis. It is essential for simplifying algebraic expressions and is used frequently in mathematics, especially when working with polynomials.

When you come across an expression like \( (a + b) \times c \), the distributive property lets you 'distribute' the \( c \) across the \( a \) and \( b \) terms, resulting in \( ac + bc \). This operation is crucial for binomial multiplication, as it enables you to expand expressions where neither factor is a monomial—just like in our exercise with \( (3x-4)(x+5) \).

Understanding the distributive property is a stepping stone to mastering more complex algebraic manipulations and is often intertwined with the FOIL method, which is a specific application of the distributive property.

The FOIL method is a mnemonic that stands for First, Outer, Inner, Last. It's a specific case of the distributive property used to multiply two binomials. This straightforward technique helps students remember the order in which to multiply the terms.

Let's break it down using our given exercise \( (3x-4)(x+5) \):

• First: Multiply the first terms in each binomial, resulting in \(3x \times x = 3x^2\).
• Outer: Multiply the outer terms, giving us \(3x \times 5 = 15x\).
• Inner: Multiply the inner terms, resulting in \(-4 \times x = -4x\).
• Last: Multiply the last terms, which is \(-4 \times 5 = -20\).

By following the FOIL sequence, multiplication of binomials becomes a more manageable process.

Once the distributive property and the FOIL method have been applied, as in our example of \( (3x-4)(x+5) \), the next step is to combine like terms. This means adding or subtracting terms that have the exact same variable raised to the same power. Combining like terms simplifies the expression and makes it easier to understand.

For instance, in our exercise, we end up with the terms \(15x\) and \(-4x\), which can be combined because they both contain the variable \(x\) to the first power. Adding them gives us \(15x - 4x = 11x\), resulting in a simplified expression of \(3x^2 + 11x - 20\).

Learning to combine like terms effectively is a key skill in algebra that enables students to simplify expressions and solve equations accurately.