We are tasked with solving for the integral $\int_{4}^{-2} (f(x) - 3) \, dx$ given that:

$\int_{-2}^{4} \frac{f(x)}{4} \, dx = -3.$

Step 1: Analyze the given integral

The integral given is:

$\int_{-2}^{4} \frac{f(x)}{4} \, dx = -3.$

We can rewrite this as:

$\frac{1}{4} \int_{-2}^{4} f(x) \, dx = -3.$

Multiplying through by 4:

$\int_{-2}^{4} f(x) \, dx = -12.$

Step 2: Use the properties of definite integrals

We need to compute:

$\int_{4}^{-2} (f(x) - 3) \, dx.$

First, use the linearity of integrals:

$\int_{4}^{-2} (f(x) - 3) \, dx = \int_{4}^{-2} f(x) \, dx - \int_{4}^{-2} 3 \, dx.$

Step 3: Flip the limits of integration

Using the property $\int_{b}^{a} g(x) \, dx = -\int_{a}^{b} g(x) \, dx$, we rewrite:

$\int_{4}^{-2} f(x) \, dx = -\int_{-2}^{4} f(x) \, dx$ and $\int_{4}^{-2} 3 \, dx = -\int_{-2}^{4} 3 \, dx.$

Thus:

$\int_{4}^{-2} (f(x) - 3) \, dx = -\int_{-2}^{4} f(x) \, dx - \left(-\int_{-2}^{4} 3 \, dx \right).$

Step 4: Substitute known values

From Step 1, we know that:

$\int_{-2}^{4} f(x) \, dx = -12.$

The second term is:

$\int_{-2}^{4} 3 \, dx = 3 \cdot (4 - (-2)) = 3 \cdot 6 = 18.$

Thus:

$\int_{4}^{-2} (f(x) - 3) \, dx = -(-12) - (-18).$

Simplify:

$\int_{4}^{-2} (f(x) - 3) \, dx = 12 + 18 = 30.$

Final Answer:

$\int_{4}^{-2} (f(x) - 3) \, dx = 30.$

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Would you like a step-by-step breakdown of any part of this solution?
Here are 5 questions to deepen your understanding:

1. How do you interpret the limits of integration when they are reversed?
2. Why does flipping the limits of integration introduce a negative sign?
3. What is the significance of the linearity of integrals in this problem?
4. How can definite integrals be used in real-world applications?
5. How do properties of definite integrals simplify calculations?

Tip: Always use properties of integrals (e.g., linearity and reversing limits) to simplify complex calculations before diving into direct computation.