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The Distributive Property is a fundamental concept in algebra. It allows you to multiply a single term by multiple terms inside parentheses. In this exercise, we used the distributive property to multiply each term inside the parentheses by the term outside, which was \(a^2\). This is how it works:

1. First, distribute \(a^2\) to each term inside the parentheses: \(a^{2}(a^{2} - 9 a - 36)\).
2. Multiply \(a^2\) by \(a^2\) to get \(a^4\).
3. Multiply \(a^2\) by \(-9a\) to get \(-9a^3\).
4. Multiply \(a^2\) by \(-36\) to get \(-36a^2\).

This property is very useful for polynomial multiplication and allows for breaking down complex expressions into simpler parts.

Exponent Rules help us handle powers of numbers and variables. When dealing with polynomial multiplication, these rules are essential. There are a few key principles to remember:

1. **Product of Powers Rule:** When you multiply two exponents with the same base, you add the exponents. For instance, \(a^2 \times a^2 = a^4\) because \(2 + 2 = 4\).
2. **Negative Exponents:** When multiplying, if one of the exponents is negative, the equation still applies. For instance, \(a^2 \times -9a = -9a^3\) because \(2 + 1 = 3\).

The main takeaway is to always add the exponents when dealing with multiplications of like bases.

Combining Like Terms is the process of simplifying expressions by merging terms that have the same variables and exponents. This is usually done after all multiplications and other operations are performed.

Here’s a step-by-step on how to combine like terms:

1. Identify terms with the same variable and exponent. In our example, we have \(a^4\), \(-9a^3\), and \(-36a^2\).
2. Add or subtract the coefficients (numbers in front of the variables). Since there are no like terms in our example, we simply list the terms together.
3. The final simplified expression is \(a^4 - 9a^3 - 36a^2\).

This process helps in creating a neat and simplified polynomial, making further calculations or interpretations easier.