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The problem asks us to find the equation of a curve whose slope is given by the derivative \( \frac{dy}{dx} = (x + 1)(x + 2) \), and which passes through the point \((-3, -\frac{3}{2})\).

### Solution:

1. **Step 1: Integrate the given derivative**  
   The equation for the slope is:
   \[
   \frac{dy}{dx} = (x + 1)(x + 2)
   \]
   Expand the product on the right-hand side:
   \[
   (x + 1)(x + 2) = x^2 + 3x + 2
   \]
   Now, integrate both sides with respect to \(x\) to find \(y(x)\):
   \[
   y = \int (x^2 + 3x + 2) \, dx
   \]
   Perform the integration:
   \[
   y = \frac{x^3}{3} + \frac{3x^2}{2} + 2x + C
   \]
   where \(C\) is the constant of integration.

2. **Step 2: Apply the initial condition**  
   We are given that the curve passes through the point \((-3, -\frac{3}{2})\). Substitute \(x = -3\) and \(y = -\frac{3}{2}\) into the equation to solve for \(C\):
   \[
   -\frac{3}{2} = \frac{(-3)^3}{3} + \frac{3(-3)^2}{2} + 2(-3) + C
   \]
   Simplify each term:
   \[
   -\frac{3}{2} = \frac{-27}{3} + \frac{3(9)}{2} + (-6) + C
   \]
   \[
   -\frac{3}{2} = -9 + \frac{27}{2} - 6 + C
   \]
   Combine the terms:
   \[
   -\frac{3}{2} = \frac{-18}{2} + \frac{27}{2} + C
   \]
   \[
   -\frac{3}{2} = \frac{9}{2} + C
   \]
   Solving for \(C\):
   \[
   C = -\frac{3}{2} - \frac{9}{2} = -\frac{12}{2} = -6
   \]

3. **Step 3: Write the final equation**  
   Now that we know \(C = -6\), the equation of the curve is:
   \[
   y = \frac{x^3}{3} + \frac{3x^2}{2} + 2x - 6
   \]

This is the equation of the curve that satisfies the given conditions.

Would you like further details or have any questions?

### Related Questions:
1. How do you determine the constant of integration for an initial value problem?
2. What is the importance of integrating to find a curve from its slope?
3. Can this method be applied to any function of \(x\) for the slope?
4. How would the solution change if the point was different?
5. How can you verify the correctness of the final equation?

**Tip:** Always remember to apply the initial conditions after integration to find the constant.

Hallar la ecuación de la curva que tiene pendiente dy/dx = (x + 1)(x + 2) y que pasa por el punto (-3, -3/2).

This problem requires finding the equation of a curve with slope dy/dx = (x + 1)(x + 2) that passes through the point (-3, -3/2). The solution involves expanding the derivative, integrating, and applying initial conditions.

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