Thank you for the clarification. The integral is:

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

Let's solve this step-by-step using substitution.

We will use the substitution:

\[ u = 3 - 4x - x^2 \]

First, differentiate \( u \) with respect to \( x \):

\[ \frac{du}{dx} = -4 - 2x \]

Thus,

\[ du = (-4 - 2x) \, dx \]

Solve for \( dx \):

\[ dx = \frac{du}{-4 - 2x} \]

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

Notice that:

\[ x + 3 \]

is already part of our substitution context. Express \( x \) in terms of \( u \):

\[ x = -2 \pm \sqrt{-1 - u} \]

However, simpler is to handle \( x + 3 \) combined:

Next integral becomes:

\[ I = \int \frac{(x + 3)\sqrt{u} \, du}{-2(x + 2)} \]

Splitting into manageable parts directly substituting often requires adjusting constants:

\[ \int u^{\frac{1}{2}} du = \int u^{\frac{1}{2}} \cdot \frac{du}{-2x-4} \]

Splitting constants simplifies:

\[ I = \int -1/2 u^{1/2} \, du \]

Solving directly:

\[ \int -1/2 u^{1/2} du = -1/3 u^{3/2} + C \]

Back-substitute \( u = 3 - 4x - x^2 \):

\[ \int (x+3)\sqrt{3-4x-x^2} \, dx = -1/3 (3-4x-x^2)^{3/2} + C \]

This represents the integral.

Would you like further details or explore related methods?

### Related Questions:
1. What are common substitutions in integration?
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4. How do constants impact the solution of integrals?
5. What are trigonometric methods for integrals?
6. How to verify integral results using differentiation?
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**Tip:** Always re-check substitution validity and constants throughout integration process ensuring clear substitutions.

Learn how to integrate (x + 3) sqrt(3 - 4x - x^2) step-by-step using the substitution method. This problem covers advanced integration techniques, including substitution and handling square roots in integrals.

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