Answer:

x = -1 and x = -3.

Step-by-step explanation:

To solve the equation |x² + 3x + 2| + x + 1 = 0, we need to consider separate cases for the expression inside the absolute value.

Case 1: (x² + 3x + 2) ≥ 0

In this case, the absolute value term simplifies to x² + 3x + 2. Therefore, the equation becomes:

(x² + 3x + 2) + x + 1 = 0

Simplifying:

x² + 4x + 3 = 0

Factoring:

(x + 1)(x + 3) = 0

Setting each factor equal to zero:

x + 1 = 0 => x = -1

x + 3 = 0 => x = -3

So, in Case 1, the solutions are x = -1 and x = -3.

Case 2: (x² + 3x + 2) < 0

In this case, the absolute value term simplifies to -(x² + 3x + 2), since the expression inside the absolute value is negative. Therefore, the equation becomes:

-(x² + 3x + 2) + x + 1 = 0

Simplifying:

-x² - 2x - 1 + x + 1 = 0

Simplifying further:

-x² - x = 0

Factoring out -x:

x(-x - 1) = 0

Setting each factor equal to zero:

x = 0 (rejected because it violates the assumption that x² + 3x + 2 < 0)

-x - 1 = 0 => x = -1

So, in Case 2, the only solution is x = -1.

Therefore, the overall solutions to the equation are x = -1 and x = -3.