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The given equation is:

\[
x(x^4 + 2x^3 + 24) + (13 + x)(1 - 2x^3) = 13
\]

Let's solve it step by step.

### Step 1: Expand both sides of the equation.

First, expand the first term:

\[
x(x^4 + 2x^3 + 24) = x^5 + 2x^4 + 24x
\]

Next, expand the second term:

\[
(13 + x)(1 - 2x^3) = 13(1 - 2x^3) + x(1 - 2x^3) = 13 - 26x^3 + x - 2x^4
\]

Now the equation becomes:

\[
x^5 + 2x^4 + 24x + (13 - 26x^3 + x - 2x^4) = 13
\]

### Step 2: Combine like terms.

Now, let's combine all the terms on the left-hand side:

\[
x^5 + (2x^4 - 2x^4) + (-26x^3) + (24x + x) + 13 = 13
\]

This simplifies to:

\[
x^5 - 26x^3 + 25x + 13 = 13
\]

### Step 3: Simplify the equation.

Subtract 13 from both sides:

\[
x^5 - 26x^3 + 25x = 0
\]

### Step 4: Factor the equation.

Factor out \( x \) from the equation:

\[
x(x^4 - 26x^2 + 25) = 0
\]

Now, we have two possible cases:

1. \( x = 0 \)
2. \( x^4 - 26x^2 + 25 = 0 \)

### Step 5: Solve \( x^4 - 26x^2 + 25 = 0 \).

Let \( y = x^2 \), so the equation becomes a quadratic in \( y \):

\[
y^2 - 26y + 25 = 0
\]

Solve this quadratic using the quadratic formula:

\[
y = \frac{-(-26) \pm \sqrt{(-26)^2 - 4(1)(25)}}{2(1)}
\]
\[
y = \frac{26 \pm \sqrt{676 - 100}}{2}
\]
\[
y = \frac{26 \pm \sqrt{576}}{2}
\]
\[
y = \frac{26 \pm 24}{2}
\]

So the two solutions for \( y \) are:

\[
y = \frac{26 + 24}{2} = 25 \quad \text{or} \quad y = \frac{26 - 24}{2} = 1
\]

### Step 6: Solve for \( x \).

Recall that \( y = x^2 \), so:

1. \( x^2 = 25 \) gives \( x = \pm 5 \).
2. \( x^2 = 1 \) gives \( x = \pm 1 \).

### Step 7: Combine all solutions.

The possible solutions for \( x \) are:

\[
x = 0, \pm 1, \pm 5
\]

### Final Answer:
The solutions to the equation are:

\[
x = 0, \pm 1, \pm 5
\]

Would you like more details on any step? Here are five related questions to explore further:

1. How do you solve quadratic equations using the quadratic formula?
2. What does factoring an equation help you achieve in algebraic problem-solving?
3. Why is it important to check solutions for higher-order polynomial equations?
4. How would you verify if all solutions are valid for the original equation?
5. What is the role of substitution in simplifying higher-order polynomials?

**Tip:** When solving equations, always check if you can factor out common terms to simplify the process early on!

Solve the equation x(x^4 + 2x^3 + 24) + (13 + x)(1 - 2x^3) = 13

This problem involves solving a polynomial equation by expanding terms, simplifying, and using the quadratic formula. The solution includes factoring and solving higher-degree polynomials, suitable for students in Grades 9-12.

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