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When we come across an algebraic expression like 5(-6x^2 - 3), the distributive property is the tool we use to simplify it. In essence, the distributive property tells us how to handle multiplication when there's a sum (or difference) inside parentheses. In more formal mathematical terms, the distributive property says that for any numbers a, b, and c, the equation a(b + c) = ab + ac holds true.

Applying the distributive property involves multiplying the number outside the parentheses (in this case, 5) by each term inside the parentheses. So, we multiply 5 by -6x^2 and by -3 separately. Here's how it looks:

• 5 * (-6x^2) = -30x^2
• 5 * (-3) = -15

By following these steps, we've used the distributive property to expand the expression and make our next steps towards simplification clearer.

Polynomial simplification is a way of reducing an algebraic expression to its simplest form. After applying the distributive property, we often end up with multiple terms that may or may not be like terms. Like terms are terms that have identical variable parts, such as 2x and 3x, and they can be combined through addition or subtraction.

In the problem 5(-6x^2 - 3), after applying the distributive property, we get -30x^2 - 15. These are not like terms because one is a term in x^2 and the other is a constant. Since they are already simplified and there are no like terms to combine, our simplification process stops here. The final expression -30x^2 - 15 is the most simplified form of the original polynomial.

Multiplication in algebra follows the same principles as multiplication with numbers, but with one key difference - we're dealing with variables as well. Variables represent unknown numbers, and when we multiply a number by a variable, the result is a term that is a product of both.

Let's take our example. When we multiply 5 by -6x^2, we multiply the coefficients (5 and -6) to get -30. However, the x^2 part remains unchanged because nothing is being multiplied with it - we're only dealing with the coefficient. Similarly, multiplying 5 by the constant -3 gives us -15, a constant term. By keeping track of both coefficients and variables during multiplication, we're able to simplify algebraic expressions correctly.