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The distributive property is a useful algebraic property that helps us multiply expressions over addition or subtraction. Here’s how it works: If we have an expression like \(a(b + c)\), the distributive property allows us to multiply each term inside the parentheses by the factor outside. Hence, \(a(b + c) = ab + ac\). In our problem, we see two cases where we apply this property:

First, we distribute \(7y^2\) in the term \(7y^2(y+2)\):

\[ 7y^2(y) + 7y^2(2) = 7y^3 + 14y^2 \]

Here, \(y^2\) was multiplied by \(y\) and \(2\) separately.

Second, we distribute \(2y\) in the term \(2y(5 + y^2)\):
\[ 2y(5) + 2y(y^2) = 10y + 2y^3 \]

This process simplifies complex expressions and sets the stage for combining like terms.

Combining like terms is an essential technique in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power.

For example, in the expression \(7y^3 + 14y^2 + 10y + 2y^3\), the like terms are:

• \(7y^3\) and \(2y^3\)
• \(14y^2\)
• \(10y\)

We group and combine these like terms by adding their coefficients. Let’s do this step-by-step:

\(7y^3 + 2y^3 = 9y^3\)
There are no other \(y^2\) terms or constant term to combine, so we write the others as they are:

\(14y^2 + 10y\)

Thus, the expression simplifies to \(9y^3 + 14y^2 + 10y\).

This process reduces the expression to its simplest form, making it easier to understand and handle in further calculations.

Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. They are an essential building block in algebra.

For example, in the problem given, the expression \(7y^2(y+2)+2y(5+y^2)\) represents a polynomial. After simplifying it step-by-step, we ended up with another polynomial: \(9y^3 + 14y^2 + 10y\).

Polynomials come in various forms:

• Monomials: Polynomials with just one term, like \(3x\) or \(-2y^2\).
• Binomials: Polynomials with two terms, like \(x + 1\) or \(4x^2 - x\).
• Trinomials: Polynomials with three terms, like \(x^2 + x + 1\).
• Higher-order Polynomials: Polynomials with more than three terms, like our result \(9y^3 + 14y^2 + 10y\).

Understanding the structure of polynomials helps in manipulating and simplifying them, as seen in our exercise. This skill is crucial for tackling algebraic equations and functions.