Let's break down the problem and find the inequality step-by-step.
1. Let [tex]\( x \)[/tex] represent the age of building [tex]\( C \)[/tex].
2. Building [tex]\( B \)[/tex] was built two years before building [tex]\( C \)[/tex]. Therefore, if [tex]\( C \)[/tex] is [tex]\( x \)[/tex] years old, building [tex]\( B \)[/tex] is [tex]\( x - 2 \)[/tex] years old.
3. Building [tex]\( D \)[/tex] was built two years before building [tex]\( B \)[/tex]. Thus, if building [tex]\( B \)[/tex] is [tex]\( x - 2 \)[/tex] years old, building [tex]\( D \)[/tex] is [tex]\( (x - 2) - 2 = x - 4 \)[/tex] years old.
4. The product of building [tex]\( B \)[/tex]'s age and building [tex]\( D \)[/tex]'s age is at least 195. Mathematically, this is represented as:
[tex]\[
(x - 2)(x - 4) \geq 195
\][/tex]
5. Now, we need to simplify the expression:
[tex]\[
(x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8
\][/tex]
6. Therefore, we have the inequality:
[tex]\[
x^2 - 6x + 8 \geq 195
\][/tex]
7. To match the inequality with the given options, rewrite it accordingly:
[tex]\[
x^2 - 6x + 8 \geq 195,
\][/tex]
So, the inequality representing this situation is:
[tex]\[
\boxed{x^2 - 6x + 8 \geq 195}
\][/tex]
Which corresponds to option C.
