What is x x(x+1)(x+3)(x+4)$$ \int $$$$  $$$$  $$

It seems like your question contains some formatting issues and might be a bit unclear. Let's break it down and clarify the components:

1. **Expression $x(x+1)(x+3)(x+4)$**:
   - This is a polynomial expression that can be expanded or analyzed.

2. **Integral $\int$**:
   - It looks like you might want to integrate a function. If you're integrating $x(x+1)(x+3)(x+4)$, we need to set up the integral properly.

Let's assume you want to integrate the polynomial $x(x+1)(x+3)(x+4)$ with respect to $x$. Here is the process:

1. **Expand the polynomial**:
   \[
   x(x+1)(x+3)(x+4) = x \cdot (x^2 + 4x + 3) \cdot (x + 4)
   \]
   \[
   = x \cdot (x^3 + 4x^2 + 3x + 4x^2 + 16x + 12)
   \]
   \[
   = x \cdot (x^3 + 8x^2 + 19x + 12)
   \]
   \[
   = x^4 + 8x^3 + 19x^2 + 12x
   \]

2. **Integrate the expanded polynomial**:
   \[
   \int (x^4 + 8x^3 + 19x^2 + 12x) \, dx
   \]
   Apply the power rule for integration:
   \[
   = \frac{x^5}{5} + \frac{8x^4}{4} + \frac{19x^3}{3} + \frac{12x^2}{2} + C
   \]
   Simplify the coefficients:
   \[
   = \frac{x^5}{5} + 2x^4 + \frac{19x^3}{3} + 6x^2 + C
   \]

So, the integral of the polynomial $x(x+1)(x+3)(x+4)$ with respect to $x$ is:
\[
\int x(x+1)(x+3)(x+4) \, dx = \frac{x^5}{5} + 2x^4 + \frac{19x^3}{3} + 6x^2 + C
\]

Do you want details or have any questions?

Here are 5 questions you might want to ask:

1. How do you expand polynomials step-by-step?
2. What is the power rule for integration?
3. How do you integrate more complex functions?
4. Can you show another example of polynomial integration?
5. How do you determine the constant of integration?

**Tip:** Always double-check your polynomial expansion before integrating to ensure accuracy.

Learn how to integrate the polynomial x(x+1)(x+3)(x+4) step-by-step with detailed explanations. This problem covers polynomial expansion and integration using the power rule, suitable for advanced high school students.

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