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The given expression is:

\[
(2x + 1)^2 - (7x + 8)^2
\]

This is a difference of squares, which can be factored using the formula:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Here, we identify \(a = 2x + 1\) and \(b = 7x + 8\). Applying the difference of squares formula:

\[
(2x + 1)^2 - (7x + 8)^2 = \left( (2x + 1) - (7x + 8) \right) \left( (2x + 1) + (7x + 8) \right)
\]

Now simplify each factor:

1. \((2x + 1) - (7x + 8) = 2x + 1 - 7x - 8 = -5x - 7\)
2. \((2x + 1) + (7x + 8) = 2x + 1 + 7x + 8 = 9x + 9\)

So, the factored form is:

\[
(2x + 1)^2 - (7x + 8)^2 = (-5x - 7)(9x + 9)
\]

This is the completely factored form of the expression.

Would you like more details or have any questions?

Here are some related questions to explore further:
1. How do we factor expressions that are not perfect squares?
2. Can the expression \((9x + 9)\) be factored further?
3. How do you recognize when to use the difference of squares formula?
4. What other factoring methods could be applied if this wasn’t a difference of squares?
5. What happens if we expand the factored expression to check our work?

**Tip:** When working with difference of squares, always check for any common factors in each binomial.

Factor completely: (2x + 1)^2 - (7x + 8)^2

Learn how to factor the expression (2x + 1)^2 - (7x + 8)^2 completely using the difference of squares formula. This step-by-step guide covers key algebraic concepts suitable for high school students in Grades 9-12.

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