In Exercises 15–58, find each product. (4x^2+5x)(4x^2−5x)

Verified step by step guidance

1. Identify the expression to be multiplied: $(4x^2+5x)(4x^2-5x)$.
2. Recognize this as a difference of squares: $(a+b)(a-b)=a^2-b^2$.
3. Assign $a=4x^2$ and $b=5x$.
4. Apply the difference of squares formula: $(4x^2{)}^2-(5x{)}^2$.
5. Simplify each term: $(4x^2{)}^2=16x^4$ and $(5x{)}^2=25x^2$.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in the first polynomial to every term in the second polynomial. This process requires applying the distributive property, ensuring that all combinations of terms are multiplied together, and then combining like terms to simplify the result.

Difference of Squares

The expression (a + b)(a - b) represents the difference of squares, which simplifies to a² - b². In the given problem, recognizing that the polynomials can be viewed in this form allows for a more straightforward calculation, as it eliminates the need for extensive distribution.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. This step is crucial after multiplying polynomials, as it helps to present the final answer in its simplest form, making it easier to interpret.