The given integral to evaluate is:

$\int_1^4 \left(4x^3 + 5x^4 + 2\right) \, dx$

Step-by-Step Solution:

1. Integrate each term:

   • For $4x^3$, the integral is: $\int 4x^3 \, dx = \frac{4x^4}{4} = x^4$
   • For $5x^4$, the integral is: $\int 5x^4 \, dx = \frac{5x^5}{5} = x^5$
   • For $2$, the integral is: $\int 2 \, dx = 2x$

   Thus, the antiderivative is: $x^4 + x^5 + 2x$

2. Evaluate the definite integral: Substitute the limits $x = 4$ and $x = 1$ into the antiderivative: $\left[x^4 + x^5 + 2x\right]_1^4$

   First, compute at $x = 4$: $(4^4) + (4^5) + 2(4) = 256 + 1024 + 8 = 1288$

   Then, compute at $x = 1$: $(1^4) + (1^5) + 2(1) = 1 + 1 + 2 = 4$

3. Subtract to find the result: $1288 - 4 = 1284$

Final Answer:

$\boxed{1284}$

Do you need further clarification or a detailed explanation? Here are some questions to expand your understanding:

1. Why do we add a constant of integration for indefinite integrals but not for definite ones?
2. How does the power rule for integration work, and when is it applied?
3. Can definite integrals be used to compute areas under curves? If so, how does this integral relate to that concept?
4. What are common mistakes to avoid when solving definite integrals?
5. How can we use software or tools to verify integrals like this one?

Tip: Always double-check the limits of integration when performing definite integrals to avoid calculation errors.