To simplify the expression $\left(\frac{3}{2}x + 1\right)^3$, follow these steps:

1. Identify the expression inside the parentheses: $\frac{3}{2}x + 1$ This expression represents a linear term where $x$ is a variable.

2. Raise the entire expression to the power of 3: $\left(\frac{3}{2}x + 1\right)^3$

   This can be expanded using the binomial theorem, which states that $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

   Here, $a = \frac{3}{2}x$ and $b = 1$.

3. Expand the expression: $\left(\frac{3}{2}x\right)^3 + 3\left(\frac{3}{2}x\right)^2(1) + 3\left(\frac{3}{2}x\right)(1^2) + 1^3$

   Now, calculate each term:

   • $\left(\frac{3}{2}x\right)^3 = \frac{27}{8}x^3$
   • $3\left(\frac{3}{2}x\right)^2(1) = 3 \times \frac{9}{4}x^2 = \frac{27}{4}x^2$
   • $3\left(\frac{3}{2}x\right)(1^2) = 3 \times \frac{3}{2}x = \frac{9}{2}x$
   • $1^3 = 1$

   Combine all the terms: $\frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1$

So, $\left(\frac{3}{2}x + 1\right)^3$ expands to $\frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1$.

Would you like more details on the expansion, or do you have any questions?

Here are some related questions you might be interested in:

1. How do you expand $(2x + 3)^3$ using the binomial theorem?
2. What is the cube of $\frac{2}{3}x + 1$?
3. How do you simplify $\left(\frac{3}{2}x - 1\right)^3$?
4. How do you find the derivative of $\left(\frac{3}{2}x + 1\right)^3$?
5. What is the expansion of $(ax + b)^3$ in general terms?

Tip: The binomial theorem is a powerful tool for expanding expressions raised to a power, especially when dealing with polynomials.