1. **State the problem:** Simplify the expression $\left(x^2 + \frac{3}{2} \times 3\right)^3$. 2. **Rewrite the expression:** The expression inside the parentheses is $x^2 + \frac{3}{2} \times 3$. 3. **Calculate the multiplication inside the parentheses:** $$\frac{3}{2} \times 3 = \frac{3 \times 3}{2} = \frac{9}{2}$$ 4. **Substitute back:** $$\left(x^2 + \frac{9}{2}\right)^3$$ 5. **Final simplified form:** The expression is now simplified to $$\left(x^2 + \frac{9}{2}\right)^3$$ This is the simplified form of the original expression. Expanding it further would involve binomial expansion, but since the problem only asks for simplification, this is the final answer.