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Understanding the distributive property is fundamental when multiplying algebraic expressions. It's a rule that enables us to multiply a single term by each term in a polynomial without changing the polynomial's value. Consider you have a number outside the parenthesis and several terms within. The distributive property allows you to 'distribute' the multiplication of that single term to all the terms inside the parenthesis.

For example, if you have the expression
\( a(b + c) \),

you can apply the distributive property like this:
\( a \times b + a \times c \).

In the context of our exercise, we multiply the monomial \(4x^2\) with each term inside the parentheses, applying this property:

Monomial multiplication refers to multiplying single-term expressions, which are monomials, with other monomials or polynomials. When you multiply monomials, the coefficients (numerical part) are multiplied together, and for any shared bases (like 'x'), you add their exponents. It's based on the exponent rule which states that when you multiply powers with the same base, you can add their exponents (\(x^a \times x^b = x^{a+b}\)).

In our original problem, \(4x^2\) is a monomial, and we multiply it by each of the monomials in the polynomial \(3x^2 - 2x + 6\). Here's how it works step by step:

• \(4x^2 \times 3x^2 = 12x^{2+2} = 12x^4\)
• \(4x^2 \times (-2x) = -8x^{2+1} = -8x^3\)
• \(4x^2 \times 6 = 24x^2\)

Once you've performed the multiplication using the distributive property and monomial multiplication rules, you'll often need to simplify the resulting expression. Simplifying helps in reducing the expression to its simplest form making it easier to read or further manipulate.

To simplify algebraic expressions, you combine like terms which are terms that have the same variables raised to the same power. It's important to note that only the coefficients of like terms are added or subtracted while the variable part remains the same.

In our problem, after multiplying, the terms \(12x^4\), \(-8x^3\), and \(24x^2\), already appear in their simplest form since there are no like terms to combine. Each term is a part of the simplified version of the original polynomial expression.