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The distributive property is a fundamental concept in algebra that allows us to simplify expressions by "distributing" multiplication over addition or subtraction inside a set of parentheses. It states that for any numbers or variables \(a\), \(b\), and \(c\), the following is true:

• \( a(b + c) = ab + ac \)
• Similarly, it applies to subtraction: \( a(b - c) = ab - ac \)

In our example, \[-6x^2(2x^2 + 3x - 5)\] we use the distributive property to multiply \(-6x^2\) with each term inside the parenthesis:

• \(-6x^2 \cdot 2x^2 \)
• \(-6x^2 \cdot 3x \)
• \(-6x^2 \cdot (-5) \)

This gives us the expression:\(-12x^4 - 18x^3 + 30x^2\), allowing us to simplify step by step before reaching the final answer.

Exponents are a shorthand way to represent repeated multiplication of a number by itself. Understanding the rules of exponents is essential when dealing with polynomial multiplication. Here are a few key exponent rules:

• Multiplying Powers with the Same Base: \(x^a \times x^b = x^{a+b}\)
• Power of a Power: \((x^a)^b = x^{a \cdot b} \)
• Power of a Product: \((ab)^n = a^n \times b^n\)

In the context of our exercise, when we multiply \(-6x^2\) and \(2x^2\), we apply the rule for multiplying powers with the same base: \[-6 \cdot 2 \times x^{2+2} = -12x^4\] We use it similarly for the other terms, ensuring the correct application of exponent rules to achieve the simplified result: \(-12x^4 - 18x^3 + 30x^2\). Each term has been evaluated by multiplying coefficients and adding the exponents of like bases.

Algebraic expressions are combinations of variables, numbers, and operations. They form the basic building blocks for equations and functions in mathematics. This exercise involves understanding and manipulating such expressions using the tools of polynomial multiplication.In the expression\(-6x^2(2x^2 + 3x - 5)\)we see an algebraic expression being multiplied by a monomial term. The challenge here involves using the distributive property to manage different terms and correctly applying exponent rules to handle the powers of variables.To work through such an expression, remember these key steps:

• Identify each term that needs to be multiplied.
• Apply the distributive property to ensure each term inside the parentheses is multiplied by the term outside.
• Simplify by applying exponent rules and combining like terms wherever possible.

These steps will enable you to break down complex algebraic expressions into simpler parts, leading to a clearer and more manageable solution.