https://file.aiquickdraw.com/mathbot/file/17309283572152va2o3p2.jpg

To add these polynomials, we’ll combine like terms step-by-step.

The given expression is:
\[
\left( \frac{1}{2}x^3 + 3x^2 + \frac{27}{4} \right) + \left( \frac{9}{7}x^3 - \frac{4}{5}x^2 - 5x \right)
\]

1. **Combine \(x^3\) terms:**
   \[
   \frac{1}{2}x^3 + \frac{9}{7}x^3 = \left(\frac{1 \cdot 7 + 9 \cdot 2}{2 \cdot 7}\right)x^3 = \frac{7 + 18}{14}x^3 = \frac{25}{14}x^3
   \]

2. **Combine \(x^2\) terms:**
   \[
   3x^2 - \frac{4}{5}x^2 = \left(\frac{3 \cdot 5 - 4 \cdot 1}{1 \cdot 5}\right)x^2 = \frac{15 - 4}{5}x^2 = \frac{11}{5}x^2
   \]

3. **Combine constant terms:**
   \[
   \frac{27}{4} + 0 = \frac{27}{4}
   \]

So, the simplified expression is:
\[
\frac{25}{14}x^3 + \frac{11}{5}x^2 - 5x + \frac{27}{4}
\]

Would you like to see detailed steps on handling the fractions, or do you have any questions?

### Related Questions
1. How would we subtract instead of adding these polynomials?
2. Can we convert each fraction to decimals for simplification?
3. How does distributing coefficients affect combining like terms?
4. What’s the process for adding polynomials with more variables?
5. How would you graph this polynomial?

### Tip
Always find a common denominator when adding or subtracting terms with fractions.

Add the polynomials: \(\left( \frac{1}{2}x^3 + 3x^2 + \frac{27}{4} \right) + \left( \frac{9}{7}x^3 - \frac{4}{5}x^2 - 5x \right)\).

Learn to add polynomials with fractional coefficients in the expression \((\frac{1}{2}x^3 + 3x^2 + \frac{27}{4}) + (\frac{9}{7}x^3 - \frac{4}{5}x^2 - 5x)\). This solution demonstrates combining like terms and using common denominators, suitable for high school algebra students.

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