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The distributive property is a fundamental aspect of algebra. It helps in expanding expressions by multiplying each term inside a set of parentheses by a factor outside of it. In the given problem, the expression is

• \((2x - 5)(-4x)\)

Here, the distributive property is applied by multiplying

• \(-4x\)

with each term inside the parentheses, which are

• \(2x\)
• \(-5\)

By using the distributive property correctly, we expand the expression to

• \(-4x \cdot 2x - 4x \cdot -5\)

This property facilitates breaking down complex algebraic expressions into simpler parts, making it easier to perform further calculations or simplifications. It's crucial to ensure every term inside the parentheses is multiplied by the term outside accurately.

Simplifying expressions involves combining like terms and performing arithmetic calculations to achieve the simplest form of the expression. After applying the distributive property to our initial problem, we arrived at

• \(-8x^2 + 20x\)

In this expression, the order of operations is adhered to, and all multiplications have been carried out. The next step in simplifying would involve combining like terms, if possible.
However, in our final expression,

• \(-8x^2\)
• \(20x\)

are not like terms because they have different powers of \(x\). Hence, no further simplification is possible. The aim of simplifying expressions is to reach a form where all components of the expression are clearly defined, and no further calculations are necessary.

Algebraic expressions describe mathematical phrases involving numbers, variables, and operation symbols. They form the backbone of algebra.
The exercise involves the multiplication of two algebraic expressions,

• \((2x - 5)\)
• \((-4x)\)

During multiplication, each term in the first expression must be multiplied by every term in the second expression. This characteristic is crucial for understanding and performing polynomial multiplication.
Each part of an algebraic expression can be broken down into terms, coefficients, and variables. For instance, in

• \(-8x^2 + 20x\)

\(-8\) and \(20\) are coefficients, whereas \(x^2\) and \(x\) are the variable parts. By grasping the components of algebraic expressions, students can manipulate and solve complex expressions more efficiently.