To solve the equation:

\[
-\frac{34 + 2x^3}{3} = 10 - x^3
\]

### Step 1: Eliminate the denominator by multiplying both sides by 3:
\[
-(34 + 2x^3) = 3(10 - x^3)
\]

This simplifies to:
\[
-(34 + 2x^3) = 30 - 3x^3
\]

### Step 2: Distribute the negative sign on the left side:
\[
-34 - 2x^3 = 30 - 3x^3
\]

### Step 3: Move all terms involving \( x^3 \) to one side and constants to the other side:
\[
-2x^3 + 3x^3 = 30 + 34
\]

This simplifies to:
\[
x^3 = 64
\]

### Step 4: Solve for \( x \) by taking the cube root of both sides:
\[
x = \sqrt[3]{64} = 4
\]

Thus, the solution is:
\[
x = 4
\]

Would you like more details on any of these steps or have any questions?

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Here are 5 related questions to consider:
1. How do you solve an equation with a higher degree polynomial?
2. What is the process for isolating variables in an equation?
3. Can cube roots have negative solutions, and how do you determine them?
4. How can you simplify equations involving fractions and polynomials?
5. What are the properties of cube roots in real and complex numbers?

**Tip:** When solving equations, always check your solution by substituting it back into the original equation.

This solution explains how to solve the polynomial equation -\frac{34 + 2x^3}{3} = 10 - x^3 step-by-step. The problem involves isolating variables, handling fractional equations, and finding cube roots. Suitable for students in Grades 10-12.

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