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The distributive property in algebra is like a magic rule that helps us simplify expressions. It states that if you have a term outside a set of parentheses, you can multiply it with each term inside the parentheses separately. This property is written mathematically as: \[ a(b + c) = ab + ac \]This means you distribute the multiplication over addition or subtraction. It's a fundamental rule that allows you to break down complex expressions into simpler parts.

• In our original exercise, we use the distributive property to multiply \( x \) with each term inside \( (4p + 5q) \).
• This results in two separate terms: one for \( 4p \) and another for \( 5q \).

As a result, we get the expression \((4p)(x) + (5q)(x)\), ready to be further simplified.

Multiplying monomials seems tricky at first, but it's all about handling the coefficients and variables in a systematic way. A monomial is a single term like \(4p\), consisting of a coefficient and variables, maybe with exponents. When you multiply two monomials, you multiply their coefficients and then their variables separately. Different rules apply:

• Multiply the coefficients (numbers in front of the variables).
• Apply the laws of exponents (i.e., add the exponents for variables that are the same).

For instance, if you're multiplying \((4p)\) with \(x\), as seen in the solution, you deal with the coefficients: \(4 \times 1 = 4\), and then write the variables next to each other: \(px\). The result is the monomial \(4px\). The same process applies when multiplying \((5q)\) with \(x\), resulting in \(5qx\). This step builds toward combining these products into a simpler form.

After performing distributive property and multiplying monomials, you'll often end up with an expression needing simplification. Simplification means making an expression as simple as possible. For our exercise, after distributing \(x\) to get \((4p)(x) + (5q)(x)\), 'simplification' means writing it clearly and cleanly: \[ 4px + 5qx \] Here, simplification involves removing brackets and combining terms truthfully:

• Each term, like \(4px\) and \(5qx\), stands alone since they contain different variable parts \(p\) and \(q\).
• The final polynomial simply lists the terms, since no like terms exist to combine further.

Expression simplification helps ensure your answer is as straightforward as possible, leaving it in its final polynomial form. This is crucial for clarity and understanding.