Step-by-step explanation:

1. x³ + 6x² + 11x + 6 = 0

To solve this equation, we can first factor out the greatest common factor of the coefficients, which is 3.

x³ + 6x² + 11x + 6 = 3(x³ + 2x² + 3x + 2) = 0

Now, we can see that (x + 2) is a factor of the left-hand side, so we can factor it out:

(x + 2)(x² + 3x + 3) = 0

Therefore, we have:

x + 2 = 0 or x² + 3x + 3 = 0

Solving for x, we get:

x = -2 or x = -1

So, the roots of the equation are x = -2 and x = -1.

2. x³ - 2x² - x + 2 = 0

To solve this equation, we can first factor out the greatest common factor of the coefficients, which is 2.

x³ - 2x² - x + 2 = 2(x² - x - 1) = 0

Now, we can see that (x - 1) is a factor of the left-hand side, so we can factor it out:

(x - 1)(x² - x + 2) = 0

Therefore, we have:

x - 1 = 0 or x² - x + 2 = 0

Solving for x, we get:

x = 1 or x = 2

So, the roots of the equation are x = 1 and x = 2.

3. x³ + 9x² + 23x + 15 = 0

To solve this equation, we can first factor out the greatest common factor of the coefficients, which is 3.

x³ + 9x² + 23x + 15 = 3(x³ + 3x² + 7x + 5) = 0

Now, we can see that (x + 5) is a factor of the left-hand side, so we can factor it out:

(x + 5)(x² + 3x + 3) = 0

Therefore, we have:

x + 5 = 0 or x² + 3x + 3 = 0

Solving for x, we get:

x = -5 or x = -3

So, the roots of the equation are x = -5 and x = -3.