Let's solve the given integral step-by-step. The integral is:

$\int \sin(4x + 3) \sqrt{3 - 4x - x^2} \, dx$

To solve this integral, we can use substitution. Let's start by setting:

$u = 3 - 4x - x^2$

Now, we need to find $du$ in terms of $dx$.

First, differentiate $u$ with respect to $x$:

$\frac{du}{dx} = -4 - 2x$

Thus,

$du = (-4 - 2x) \, dx$

Let's solve for $dx$:

$dx = \frac{du}{-4 - 2x}$

Now, we need to express $\sin(4x + 3)$ and $\sqrt{3 - 4x - x^2}$ in terms of $u$.

From our substitution:

$\sqrt{3 - 4x - x^2} = \sqrt{u}$

Next, let's express $\sin(4x + 3)$ in terms of $u$. Notice that we have $4x$ inside the sine function. To make things simpler, let's consider if there's any trigonometric identity or further substitution that can simplify this expression, but often this may require recognizing a pattern or using complex functions, which isn't immediately apparent here.

This integral involves non-trivial functions, suggesting the need for more advanced techniques or even numerical methods for exact evaluation. Let's step back and reconsider the substitution $t = 2 + x$:

1. Set $t = 2 + x$, then $dt = dx$.
2. Adjust limits if definite integration or update functions.

$\int \sin(4x + 3) \sqrt{3 - 4x - x^2} \, dx$

can be recalculated, often found in advanced integral tables or computer algebra systems for a closed form.

For now, recognize more advanced techniques may be required, and exploring further into differential equations or approximation methods, or numerical integration like Simpson's Rule, may become necessary.

Would you like to explore those advanced methods or related topics further?

Here are 8 related questions for exploration:

1. What is substitution in integration, and when is it used?
2. How can trigonometric identities simplify integration problems?
3. What are common techniques for integrating functions involving square roots?
4. How do you handle integrals with non-trivial functions?
5. What are advanced methods for solving difficult integrals?
6. Can numerical methods approximate this integral effectively?
7. How does changing variables simplify integral problems?
8. Are there specific tables for complex integral solutions?

Tip: When dealing with complex integrals, sometimes re-evaluating your substitution or transformation approach can offer new insights and simplifications.