We need to solve the equation:

\[
|x + 1| = x^2 - 11
\]

### Step 1: Consider the definition of the absolute value.
The equation \( |x + 1| = x^2 - 11 \) can be split into two cases:

1. **Case 1: \( x + 1 \geq 0 \)**
   \[
   x + 1 = x^2 - 11
   \]
   Simplifying:
   \[
   x^2 - x - 12 = 0
   \]

2. **Case 2: \( x + 1 < 0 \)**
   \[
   -(x + 1) = x^2 - 11
   \]
   Simplifying:
   \[
   x^2 + x - 10 = 0
   \]

### Step 2: Solve each quadratic equation.

#### Case 1: Solve \( x^2 - x - 12 = 0 \)

Factor the quadratic equation:
\[
x^2 - x - 12 = (x - 4)(x + 3) = 0
\]

So, the solutions are:
\[
x = 4 \quad \text{or} \quad x = -3
\]

Check these solutions against the condition \( x + 1 \geq 0 \):
- For \( x = 4 \): \( x + 1 = 5 \geq 0 \) (valid)
- For \( x = -3 \): \( x + 1 = -2 \geq 0 \) (not valid)

So, the valid solution from Case 1 is \( x = 4 \).

#### Case 2: Solve \( x^2 + x - 10 = 0 \)

Factor or use the quadratic formula:
\[
x = \frac{-1 \pm \sqrt{1 + 40}}{2} = \frac{-1 \pm \sqrt{41}}{2}
\]

This yields two solutions:
\[
x = \frac{-1 + \sqrt{41}}{2} \quad \text{or} \quad x = \frac{-1 - \sqrt{41}}{2}
\]

Check these solutions against the condition \( x + 1 < 0 \):
- For \( x = \frac{-1 + \sqrt{41}}{2} \): This value is positive, so it does not satisfy \( x + 1 < 0 \).
- For \( x = \frac{-1 - \sqrt{41}}{2} \): This value is negative, so it does satisfy \( x + 1 < 0 \).

Thus, the valid solution from Case 2 is \( x = \frac{-1 - \sqrt{41}}{2} \).

### Step 3: Conclusion

The two valid solutions to the equation are:

\[
\boxed{x = 4 \text{ or } x = \frac{-1 - \sqrt{41}}{2}}
\]

Would you like more details or have any questions? Here are some related problems:

1. Solve \( |2x - 5| = x^2 + 2x - 15 \).
2. Find the solutions to \( |x + 3| = x^2 - 9 \).
3. Solve \( |x - 2| = 2x^2 + x - 8 \).
4. Determine the roots of \( |x - 4| = x^2 - 16 \).
5. Solve the quadratic equation \( x^2 + 3x - 10 = 0 \).

**Tip:** When dealing with absolute value equations, always consider the different cases based on the definition of the absolute value function.

Learn how to solve the absolute value equation |x + 1| = x^2 - 11 with a detailed step-by-step solution. This problem covers key concepts in absolute value equations and quadratic equations, including the quadratic equation formula. Suitable for advanced high school students (Grades 10-12).

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