how do you factor and simplify, 3 cot^2 x + 6 cot x + 3? please help!?

Let u = cot x. Then we get this:

3 cot² x + 6 cot x + 3 =

3u² + 6u + 3 = 3(u² + 2u + 1) =

3(u + 1)².

Substituting cot x back in for u, we get this:

3(u + 1)² = 3(cot x + 1)².

The solution is correct. We can check it by using x = π/4 = 45°.

First we use the simplified expression:

tan 45° = cot 45° = 1

3(cot 45° + 1)² = 3(1 + 1)² = 3(2)² = 3(4) = 12.

Now we use the original expression:

3 cot² 45° + 6 cot 45° + 3 = 3(1)² + 6(1) + 3 = 3 + 6 + 3 = 12.

  set: u = cotx

 = 3u² + 6u + 3

 = 3(u + 1)²

 = 3(cotx + 1)²