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The Distributive Property is an essential concept in algebra that allows you to multiply a single term and two or more terms inside a parenthesis, separately. In other words, you distribute the multiplication over addition or subtraction.
In the context of binomials, we use this property extensively when multiplying. For example, with the binomials \( (x-8)(x+9) \), we can think of multiplying each term in the first binomial by every term in the second binomial.
This lays the foundation for using the FOIL method, which is specifically designed for multiplying two binomials. The step-by-step nature of the Distributive Property makes it a reliable way to manage larger and more complex multiplications.
To sum up, you are essentially breaking down the multiplication into smaller, simpler parts and then combining the results to get the final product.

The FOIL Method is a shortcut specifically used for multiplying two binomials.
FOIL stands for:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms in the binomials.
• Inside: Multiply the inner terms in the binomials.
• Last: Multiply the last terms in each binomial.

Using the binomials \( (x - 8)(x + 9) \), the FOIL method steps are:

• \textbf{First:} \( x \times x = x^2 \)
• \textbf{Outside:} \( x \times 9 = 9x \)
• \textbf{Inside:} \( -8 \times x = -8x \)
• \textbf{Last:} \( -8 \times 9 = -72 \)

After performing these multiplications, you get \( x^2 + 9x - 8x - 72 \). The FOIL method helps break down the problem into manageable steps, making it easier to track each part of the multiplication process.

After using the FOIL method to expand the binomials, the next crucial step is to combine like terms. Like terms are terms that contain the same variable raised to the same power.
In our example, after using the FOIL method, we get \( x^2 + 9x - 8x - 72 \). Here, \( 9x \) and \( -8x \) are like terms because they both contain the variable \( x \).
To combine them, you simply add or subtract their coefficients (the numerical part). So, \( 9x - 8x = x \).
Thus, we can simplify our expression to \( x^2 + x - 72 \).
Combining like terms simplifies the expression, making it cleaner and easier to understand. This step is essential for finalizing your solution and ensuring it's in its simplest form.