Understanding the Equation
To solve the equation \( x^5 - 2\left(\frac{x}{3} + 1\right) = \frac{2x}{3} + 3 - 11x \), we need to simplify and rearrange the terms.

Step 1: Expand and Rearrange
- Expand the left side:
- \( x^5 - \frac{2x}{3} - 2 \)
- The right side simplifies to:
- \( \frac{2x}{3} + 3 - 11x = \frac{2x}{3} - \frac{33x}{3} + 3 = -\frac{31x}{3} + 3 \)
Now the equation looks like this:
\( x^5 - \frac{2x}{3} - 2 = -\frac{31x}{3} + 3 \)

Step 2: Combine Like Terms
- Move all terms to one side:
\( x^5 - \frac{2x}{3} + \frac{31x}{3} - 2 - 3 = 0 \)
This simplifies to:
\( x^5 + \frac{29x}{3} - 5 = 0 \)

Step 3: Multiply by 3 to Eliminate the Fraction
To make calculations easier, multiply the entire equation by 3:
\( 3x^5 + 29x - 15 = 0 \)

Step 4: Finding the Roots
Now, solving \( 3x^5 + 29x - 15 = 0 \) requires techniques such as synthetic division or numerical methods.

Conclusion
The polynomial equation can be solved using various methods, but it's crucial to check for rational roots or use graphing calculators for approximate solutions.
In summary, the equation transforms into a polynomial that can be analyzed for its roots to find the values of \( x \).