To solve the equation \( x|x+1| + 1 = 0 \), we will consider two cases based on the definition of the absolute value function.
### Step 1: Identify cases for the absolute value
The expression \( |x + 1| \) can be defined in two cases:
1. **Case 1:** \( x + 1 \geq 0 \) (i.e., \( x \geq -1 \))
2. **Case 2:** \( x + 1 < 0 \) (i.e., \( x < -1 \))
### Step 2: Solve Case 1 (\( x \geq -1 \))
In this case, \( |x + 1| = x + 1 \). Therefore, the equation becomes:
\[
x(x + 1) + 1 = 0
\]
This simplifies to:
\[
x^2 + x + 1 = 0
\]
Now, we can use the quadratic formula to find the solutions:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1, b = 1, c = 1 \):
\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2}
\]
This results in:
\[
x = \frac{-1 \pm i\sqrt{3}}{2}
\]
Since these solutions are complex, they do not satisfy the condition \( x \geq -1 \).
### Step 3: Solve Case 2 (\( x < -1 \))
In this case, \( |x + 1| = -(x + 1) \). Therefore, the equation becomes:
\[
x(- (x + 1)) + 1 = 0
\]
This simplifies to:
\[
-x^2 - x + 1 = 0 \quad \Rightarrow \quad x^2 + x - 1 = 0
\]
Again, we apply the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1, b = 1, c = -1 \):
\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2}
\]
Now, we have two potential solutions:
\[
x = \frac{-1 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{5}}{2}
\]
We need to check if these solutions satisfy the condition \( x < -1 \):
- For \( x = \frac{-1 - \sqrt{5}}{2} \): This value is definitely less than -1.
- For \( x = \frac{-1 + \sqrt{5}}{2} \): We need to check if this is less than -1. Since \( \sqrt{5} \approx 2.236 \), we have:
\[
\frac{-1 + 2.236}{2} \approx \frac{1.236}{2} \approx 0.618 \quad \text{(which is not less than -1)}
\]
Thus, the only valid solution from Case 2 is:
\[
x = \frac{-1 - \sqrt{5}}{2}
\]
### Final Solutions
The solutions of the equation \( x|x+1| + 1 = 0 \) are:
\[
x = \frac{-1 - \sqrt{5}}{2}
\]
