The given expression is the integral:

\[
\int x(x - 3)(x + 5) \, dx
\]

To solve this integral, we'll first expand the polynomial \(x(x - 3)(x + 5)\) and then integrate the resulting polynomial term by term.

### Step 1: Expand the Polynomial
Let's expand the expression \(x(x - 3)(x + 5)\).

1. First, expand \((x - 3)(x + 5)\):
   \[
   (x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15
   \]
   
2. Now, multiply by \(x\):
   \[
   x(x^2 + 2x - 15) = x^3 + 2x^2 - 15x
   \]

So, the integrand becomes \(x^3 + 2x^2 - 15x\).

### Step 2: Integrate Term by Term
Now, integrate the polynomial \(x^3 + 2x^2 - 15x\) with respect to \(x\):

\[
\int (x^3 + 2x^2 - 15x) \, dx = \frac{x^4}{4} + \frac{2x^3}{3} - \frac{15x^2}{2} + C
\]

Here, \(C\) is the constant of integration.

### Final Answer
The integral of \(x(x - 3)(x + 5)\) with respect to \(x\) is:

\[
\frac{x^4}{4} + \frac{2x^3}{3} - \frac{15x^2}{2} + C
\]

Would you like further details on any of the steps?

### Related Questions:
1. How do you expand products of polynomials in general?
2. Can you integrate more complex polynomials using similar methods?
3. What are the rules for integrating power functions?
4. How does integration work with different types of functions, like trigonometric or exponential?
5. What are some common mistakes to avoid when expanding and integrating polynomials?

**Tip:** When integrating polynomials, always check that each term is correctly expanded before performing the integration step.

Learn how to integrate the polynomial x(x - 3)(x + 5) step-by-step. This problem covers the integration of polynomial functions, expanding polynomials, and applying the power rule for integration. Suitable for advanced high school students.

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