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------------------------------------------------ Select the correct answer.

A construction company is analyzing which of its older projects need renovation. Building B was built two years before building C. Building D was built two years before building B. The product of building B's age and building D's age is at least 195. If [tex]x[/tex] represents the age of building C, which inequality represents this situation?

A. [tex]x^2 + 4 \geq 195[/tex]
B. [tex]x^2 + 8x + 16 \geq 195[/tex]
C. [tex]x^2 + 6x + 8 \geq 195[/tex]
D. [tex]x^2 + 4x + 4 \geq 195[/tex]

Let's break down the problem to find the correct inequality:

1. Understanding the age differences:
- Let [tex]\( x \)[/tex] represent the age of building C.
- Building B was built two years before building C, so its age is [tex]\( x + 2 \)[/tex].
- Building D was built two years before building B, so its age is [tex]\( (x + 2) + 2 = x + 4 \)[/tex].

2. Set up the inequality for the product of building ages:
- The product of the ages of Building B and Building D must be at least 195:
[tex]\[
(x + 2)(x + 4) \geq 195
\][/tex]

3. Expand and simplify the expression:
- First, expand the left-hand side:
[tex]\[
(x + 2)(x + 4) = x(x + 4) + 2(x + 4) = x^2 + 4x + 2x + 8
\][/tex]
- Combine the like terms:
[tex]\[
x^2 + 6x + 8
\][/tex]

4. Formulate the inequality:
- Now, place the expanded expression into the inequality setup:
[tex]\[
x^2 + 6x + 8 \geq 195
\][/tex]

The correct inequality that represents this situation is:

C. [tex]\( x^2 + 6x + 8 \geq 195 \)[/tex]

This matches choice C from the given options.