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The distributive property is a fundamental concept in algebra that allows us to simplify and solve equations more easily. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. In algebraic terms, this property is expressed as \( a(b + c) = ab + ac \).
In the given exercise, the distributive property is applied to expand the binomials. We start with \( (a+9)(a+4) \). To apply the distributive property, distribute each term of the first binomial to each term of the second binomial:

• First, distribute \(a\): \( a(a+4) = a^2 + 4a \).
• Next, distribute \(9\): \( 9(a+4) = 9a + 36 \).

Combining these results, we get \( a^2 + 4a + 9a + 36 \).
This method is crucial because it transforms complex expressions into a simpler, more manageable form for further operations.

Expanding binomials involves using the distributive property to multiply two binomials together. This method is often remembered by the FOIL (First, Outer, Inner, Last) method, which stands for the order in which you multiply the terms:

• First: Multiply the first terms of each binomial. In \( (a+9)(a+4) \), we get \( a \times a = a^2 \).
• Outer: Multiply the outer terms. Here, that's \( a \times 4 = 4a \).
• Inner: Multiply the inner terms, \( 9 \times a = 9a \).
• Last: Multiply the last terms, \( 9 \times 4 = 36 \).

Combining all these products, we get \( a^2 + 4a + 9a + 36 \). Expanding binomials is a critical skill for simplifying and solving polynomial expressions.

Combining like terms is the process of simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. This is essential for simplifying polynomial expressions.
In our exercise, after expanding the binomials from \( (a+9)(a+4) \), we get the expression \( a^2 + 4a + 9a + 36 \). Here, \( 4a \) and \( 9a \) are like terms because they have the same variable raised to the same power (\(a\)):

• Combine \( 4a \) and \( 9a \) to get \( 13a \).

So, the simplified polynomial is \( a^2 + 13a + 36 \). This final expression represents the polynomial that can be factored back into the original binomials \( (a+9)(a+4) \). Combining like terms helps streamline polynomial expressions and prepares them for further operations or factoring.