Find each product. See Example 5. (4x² - 5y) (4x² + 5y)

Verified step by step guidance

1. Recognize the expression as a difference of squares: $(a-b)(a+b)=a^2-b^2$.
2. Identify $a$ and $b$ in the expression: $a=4x^2$ and $b=5y$.
3. Apply the difference of squares formula: $(4x^2{)}^2-(5y{)}^2$.
4. Calculate $(4x^2{)}^2$ which is $16x^4$.
5. Calculate $(5y{)}^2$ which is $25y^2$.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity that states that the product of two binomials in the form (a - b)(a + b) equals a² - b². This concept is crucial for simplifying expressions where one binomial is the negative of the other, allowing for quick factorization and simplification.

Binomial Multiplication

Binomial multiplication involves multiplying two binomials, which can be done using the distributive property or the FOIL method (First, Outside, Inside, Last). Understanding this process is essential for expanding expressions like (4x² - 5y)(4x² + 5y) and accurately combining like terms.

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 concept is important in the final steps of polynomial multiplication, as it helps to consolidate the expression into its simplest form, making it easier to interpret and use.