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The problem asks for the area of the shaded region between the curve \( f(x) = 10x + 3x^2 - x^3 \) and the line \( g(x) = 0 \) (which represents the x-axis).

### Steps to find the area:

1. **Find the points of intersection:**
   Since \( g(x) = 0 \) is the x-axis, the points where \( f(x) = 0 \) will be the bounds of the shaded region. Solve \( f(x) = 10x + 3x^2 - x^3 = 0 \) to find these points.

2. **Solve the cubic equation:**
   \[
   x^3 - 3x^2 - 10x = 0
   \]
   Factor the equation:
   \[
   x(x^2 - 3x - 10) = 0
   \]
   Now solve for \( x \):
   - \( x = 0 \) (one solution)
   - Solve \( x^2 - 3x - 10 = 0 \) using the quadratic formula:
     \[
     x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}
     \]
     \[
     x = \frac{3 \pm \sqrt{9 + 40}}{2}
     \]
     \[
     x = \frac{3 \pm \sqrt{49}}{2}
     \]
     \[
     x = \frac{3 \pm 7}{2}
     \]
     This gives two solutions: \( x = 5 \) and \( x = -2 \).

   Therefore, the points of intersection are \( x = -2 \), \( x = 0 \), and \( x = 5 \).

3. **Set up the integral:**
   To find the area between the curve and the x-axis, compute the definite integral of \( f(x) \) from \( x = -2 \) to \( x = 5 \).

   The area \( A \) is:
   \[
   A = \int_{-2}^{5} (10x + 3x^2 - x^3) \, dx
   \]

4. **Evaluate the integral:**
   First, find the antiderivative of \( f(x) \):
   \[
   \int (10x + 3x^2 - x^3) \, dx = 5x^2 + x^3 - \frac{x^4}{4} + C
   \]
   Now, evaluate from \( x = -2 \) to \( x = 5 \):
   \[
   A = \left[ 5x^2 + x^3 - \frac{x^4}{4} \right]_{-2}^{5}
   \]
   Substituting the limits:

   For \( x = 5 \):
   \[
   5(5^2) + (5^3) - \frac{5^4}{4} = 5(25) + 125 - \frac{625}{4} = 125 + 125 - 156.25 = 93.75
   \]

   For \( x = -2 \):
   \[
   5(-2^2) + (-2^3) - \frac{(-2)^4}{4} = 5(4) - 8 - \frac{16}{4} = 20 - 8 - 4 = 8
   \]

   Now subtract the results:
   \[
   A = 93.75 - 8 = 85.75
   \]

Thus, the area of the shaded region is **85.75** or \( \frac{343}{4} \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the area between two curves?
2. What is the geometric interpretation of definite integrals?
3. Can cubic functions have more than two turning points?
4. How do you solve cubic equations using factoring?
5. What are the properties of cubic functions in terms of symmetry?

**Tip:** Always sketch the curve and the region of interest to get a visual understanding of the problem before setting up integrals.

Find the area of the shaded region for the function f(x) = 10x + 3x^2 - x^3 and the line g(x) = 0.

This problem asks to find the area of the shaded region between the cubic function f(x) = 10x + 3x^2 - x^3 and the x-axis. The solution involves using integral calculus and factoring to find the points of intersection and evaluating the definite integral.

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