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The distributive property is a fundamental concept in algebra that allows you to multiply a single term by each term within a set of parentheses. When working with binomials, the distributive property involves expanding the expression by distributing each term across the other.

In the context of multiplying binomials like \text{(u-5)(u-9)}, you distribute each term in the first binomial across each term in the second binomial. This results in multiplying the first term by both terms in the second binomial, and then multiplying the second term by both terms in the second binomial.

Here's a step-by-step breakdown to illustrate:

• First: Multiply the first terms: \(u \times u = u^2\)
• Outer: Multiply the outer terms: \(u \times -9 = -9u\)
• Inner: Multiply the inner terms: \(-5 \times u = -5u\)
• Last: Multiply the last terms: \(-5 \times -9 = 45\)

Combining these using the distributive property gives the intermediate expression: \(u^2 - 9u - 5u + 45\).

The FOIL method is a shortcut for applying the distributive property specifically to binomials. FOIL stands for First, Outer, Inner, Last, which refers to the order in which you multiply the terms of each binomial.

Using the FOIL method for \text{(u-5)(u-9)}, we can easily remember the steps:

• First: Multiply the first terms: \(u \times u = u^2\)
• Outer: Multiply the outer terms: \(u \times -9 = -9u\)
• Inner: Multiply the inner terms: \(-5 \times u = -5u\)
• Last: Multiply the last terms: \(-5 \times -9 = 45\)

This method simplifies the distribution process and ensures all products are covered, leading to the same intermediate result: \(u^2 - 9u - 5u + 45\).

After using the distributive property or the FOIL method, you often end up with several terms that need simplification. These terms can sometimes be combined if they are 'like terms'—terms that have the same variable raised to the same power.

In our example, the terms \(-9u\) and \(-5u\) are like terms because they both contain the variable \(u\) raised to the first power. We can combine them by adding their coefficients:

\(-9u - 5u = -14u\)

So, combining the like terms gives us:

• \(u^2\) (the term from the First multiplication)
• \(-14u\) (the combined Outer and Inner terms)
• \(45\) (the term from the Last multiplication)

The final expression is therefore:

\(u^2 - 14u + 45\)