Find each product. See Examples 5 and 6. [(3q+5)-p][(3q+5)+p]

Verified step by step guidance

1. Recognize that the expression $[(3q+5)-p][(3q+5)+p]$ is in the form of a difference of squares: $(a-b)(a+b)=a^2-b^2$.
2. Identify $a$ as $3q+5$ and $b$ as $p$.
3. Apply the difference of squares formula: $(3q+5{)}^2-p^2$.
4. Expand $(3q+5{)}^2$ using the formula $(x+y{)}^2=x^2+2xy+y^2$, where $x=3q$ and $y=5$.
5. Combine the expanded form of $(3q+5{)}^2$ with $-p^2$ to express the final expanded product.

Key Concepts

Factoring and Expanding

Factoring and expanding are fundamental algebraic techniques used to manipulate expressions. Factoring involves rewriting an expression as a product of its factors, while expanding involves distributing terms to eliminate parentheses. Understanding these processes is crucial for simplifying expressions and solving equations.

Difference of Squares

The difference of squares is a specific algebraic identity that states that the product of two binomials in the form (a - b)(a + b) equals a² - b². This identity is essential for simplifying expressions that involve the subtraction and addition of the same terms, making it easier to find products like the one in the question.

Binomial Products

Binomial products refer to the multiplication of two binomials, which can be approached using the distributive property or special identities. Recognizing patterns in binomial products, such as the difference of squares or the square of a binomial, helps in efficiently calculating the result and simplifying the expression.