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When you come across a multiplication problem that involves parentheses, you often need to apply the distributive property. This property is a foundational concept in algebra that allows you to multiply a number outside the parentheses by each term inside the parentheses.
This way, you distribute the multiplication over addition or subtraction. In our case, we have:

• A numerical constant, which is 5
• Terms inside the parentheses: \(2x^2\), \(-4xy\), and \(-y^2\)

Applying the distributive property has helped us break the problem into smaller, more manageable parts, multiplying 5 by each term individually. This gives:

• \(5 \times 2x^2 = 10x^2\)
• \(5 \times -4xy = -20xy\)
• \(5 \times -y^2 = -5y^2\)

Through this property, you ensure that each term inside the parentheses is multiplied accordingly, allowing for a more simplified expression.

Polynomials are expressions that consist of variables raised to whole number powers and their coefficients, combined using addition or subtraction. In simpler terms, they are mathematical sentences composed of terms. In the exercise, the polynomial inside the parentheses is:

• \(2x^2\)
• \(-4xy\)
• \(-y^2\)

Each term is a piece of the polynomial puzzle:

• \(2x^2\): Here, \(2\) is the coefficient, and \(x^2\) indicates the variable \(x\) raised to the power of 2.
• \(-4xy\): A coefficient of \(-4\), and it involves two variables, \(x\) and \(y\), multiplied together.
• \(-y^2\): A single term with \(y\) raised to the power of 2, with an implied coefficient of \(-1\).

By understanding these components, you can better comprehend how each part interacts, especially when the distributive property is applied, as seen when simplifying algebraic expressions.

Algebraic manipulation involves using mathematical operations to transform expressions into simpler or alternative forms. This is particularly useful for solving equations or simplifying complex expressions. Our problem required executing the distributive property—a form of manipulation—to combine and simplify the terms.
After distributing the 5 over the polynomial, we arrived at the expression:\(10x^2 - 20xy - 5y^2\). Now, this is a simplified expression where each term has been multiplied by the constant and any original operations, like addition or subtraction, have been considered.
Algebraic manipulation often requires identifying and applying familiar rules or strategies, such as:

• Combining like terms
• Expanding expressions
• Factoring

By practicing these techniques, you can become adept at simplifying complex algebraic structures and obtaining more workable expressions or solutions.