To evaluate the definite integral of $10 + 4x - 3x^2$ from $x = 2$ to $x = 4$, we proceed as follows:

Step 1: Find the indefinite integral

The integral of $10 + 4x - 3x^2$ with respect to $x$ is: $\int (10 + 4x - 3x^2) \, dx = 10x + 2x^2 - x^3 + C$

Step 2: Apply the limits

The definite integral is: $\int_{2}^{4} (10 + 4x - 3x^2) \, dx = \left[ 10x + 2x^2 - x^3 \right]_2^4$

First, calculate the expression at $x = 4$: $\left(10(4) + 2(4)^2 - (4)^3\right) = 40 + 32 - 64 = 8$

Next, calculate the expression at $x = 2$: $\left(10(2) + 2(2)^2 - (2)^3\right) = 20 + 8 - 8 = 20$

Step 3: Subtract the values

$\int_{2}^{4} (10 + 4x - 3x^2) \, dx = 8 - 20 = -12$

Final Answer:

$\boxed{-12}$

Would you like me to explain any specific steps or evaluate another similar problem? Here are some related questions you might consider:

1. How is the power rule for integration derived?
2. What are some applications of definite integrals in real-world problems?
3. Can the integral of this polynomial be evaluated using numerical methods for verification?
4. How does the integral change if the bounds are reversed (from $4$ to $2$)?
5. What happens if additional terms are added to the integrand, such as a trigonometric function?

Tip: Always check your limits of integration carefully, as swapping them changes the sign of the result.