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Let's solve the given equation step by step. The equation is:

\[
\left( x^2 + 2x \right)^2 - 11 \left( x^2 + 2x \right) + 24 = 0
\]

### Step 1: Substitution

To simplify this, we can use substitution. Let:
\[
u = x^2 + 2x
\]

Substituting \( u \) into the equation, we get:
\[
u^2 - 11u + 24 = 0
\]

### Step 2: Solving the quadratic equation

This is a quadratic equation, and we can solve it using factoring or the quadratic formula.

First, try factoring:
\[
u^2 - 11u + 24 = (u - 8)(u - 3) = 0
\]

So, the solutions are:
\[
u = 8 \quad \text{or} \quad u = 3
\]

### Step 3: Back-substitution

Now, we need to substitute back \( u = x^2 + 2x \). We have two cases to solve:

1. \( x^2 + 2x = 8 \)
2. \( x^2 + 2x = 3 \)

#### Case 1: \( x^2 + 2x = 8 \)

Rearrange the equation:
\[
x^2 + 2x - 8 = 0
\]

Now, solve this quadratic equation. Using the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-8)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 32}}{2} = \frac{-2 \pm \sqrt{36}}{2} = \frac{-2 \pm 6}{2}
\]
This gives two solutions:
\[
x = \frac{-2 + 6}{2} = 2 \quad \text{or} \quad x = \frac{-2 - 6}{2} = -4
\]

#### Case 2: \( x^2 + 2x = 3 \)

Rearrange the equation:
\[
x^2 + 2x - 3 = 0
\]

Again, use the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-3)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}
\]
This gives two solutions:
\[
x = \frac{-2 + 4}{2} = 1 \quad \text{or} \quad x = \frac{-2 - 4}{2} = -3
\]

### Final Solution Set:
The solution set is:
\[
x = 2, -4, 1, -3
\]

Thus, the correct answer for the solution set is: **\( \{2, -4, 1, -3\} \)**.

Would you like a more detailed explanation on any step? 

Here are five related questions:
1. How do you factor quadratic equations?
2. What is the quadratic formula, and how is it derived?
3. Can all quadratic equations be solved by factoring?
4. How can we verify if the solutions are correct by substituting them back into the original equation?
5. How can substitution simplify more complex equations?

**Tip**: Always check your solutions by substituting them back into the original equation to ensure they satisfy it!

(x^2 + 2x)^2 - 11(x^2 + 2x) + 24 = 0. Solve using substitution.

Learn how to solve the equation (x^2 + 2x)^2 - 11(x^2 + 2x) + 24 = 0 by using substitution and quadratic methods. This step-by-step solution covers key algebraic concepts such as substitution, factoring, and the quadratic formula, perfect for students in Grades 9-11.

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