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

To solve the problem, let's go through it step-by-step:

1. Understand the Variables:
- Let [tex]\( x \)[/tex] represent the age of building C.

2. Determine the Ages of Buildings B and D:
- 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]\( x + 4 \)[/tex].

3. Set Up the Inequality:
- We know the product of the ages of buildings B and D must be at least 195, which leads to the inequality:
[tex]\[
(x + 2) \times (x + 4) \geq 195
\][/tex]

4. Expand the Expression:
- Expanding [tex]\((x + 2) \times (x + 4)\)[/tex], we have:
[tex]\[
x^2 + 4x + 2x + 8 = x^2 + 6x + 8
\][/tex]

5. Form the Inequality:
- Substitute the expanded expression back into the inequality:
[tex]\[
x^2 + 6x + 8 \geq 195
\][/tex]

6. Identify the Correct Answer:
- The inequality [tex]\( x^2 + 6x + 8 \geq 195 \)[/tex] matches option C in the given choices.

Therefore, the correct option that represents the situation is:

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