To solve the equation \( |x^2 + 2024| = 2024 |x + 1| \), we start by considering the different cases that arise from the absolute values.

### Case 1: \( x^2 + 2024 \geq 0 \) and \( x + 1 \geq 0 \)

In this case, the equation becomes:

\[
x^2 + 2024 = 2024 (x + 1)
\]

Simplifying this, we get:

\[
x^2 + 2024 = 2024x + 2024
\]

Subtracting \( 2024x + 2024 \) from both sides yields:

\[
x^2 - 2024x = 0
\]

Factoring gives:

\[
x(x - 2024) = 0
\]

Thus, the solutions are:

\[
x = 0 \quad \text{or} \quad x = 2024
\]

We need to check if these solutions satisfy the conditions of this case:
- For \( x = 0 \): \( 0^2 + 2024 = 2024 \geq 0 \) and \( 0 + 1 = 1 \geq 0 \) (valid).
- For \( x = 2024 \): \( 2024^2 + 2024 \geq 0 \) (valid) and \( 2024 + 1 = 2025 \geq 0 \) (valid).

Both solutions are valid.

### Case 2: \( x^2 + 2024 \geq 0 \) and \( x + 1 < 0 \)

Here, we have:

\[
x^2 + 2024 = -2024(x + 1)
\]

This simplifies to:

\[
x^2 + 2024 = -2024x - 2024
\]

Rearranging gives:

\[
x^2 + 2024x + 4048 = 0
\]

We find the discriminant \( D \):

\[
D = 2024^2 - 4 \cdot 1 \cdot 4048 = 4094976 - 16192 = 4094784
\]

Calculating the roots:

\[
x = \frac{-2024 \pm \sqrt{4094784}}{2} = \frac{-2024 \pm 2024}{2}
\]

The roots are:

\[
x = 0 \quad \text{and} \quad x = -2024
\]

We check these against the case conditions:
- For \( x = 0 \): Valid in case 1.
- For \( x = -2024 \): \(-2024 \geq -1\) (valid) and \( |-2024| + 2024 \geq 0 \).

### Case 3: \( x^2 + 2024 < 0 \)

Since \( x^2 + 2024 \) is always positive (as \( 2024 > 0 \)), this case is impossible and thus no solutions arise.

### Conclusion

The valid solutions are:

\[
x = 0, \quad x = 2024, \quad x = -2024
\]

Thus, the final set of solutions is \( x = 0, 2024, -2024 \).