Let's work through the problem step by step.

Part a) Partial Fraction Decomposition

We are asked to decompose the following rational function into simpler fractions:

$\frac{7x^4 + 44x^2 - 8x + 63}{(x - 4)(x^2 + 3)^2}$

Steps:

1. Set up the general form of the decomposition:

   Since the denominator consists of two factors, $(x - 4)$ and $(x^2 + 3)^2$, we write the partial fraction decomposition as:

   $\frac{7x^4 + 44x^2 - 8x + 63}{(x - 4)(x^2 + 3)^2} = \frac{A}{x - 4} + \frac{Bx + C}{x^2 + 3} + \frac{Dx + E}{(x^2 + 3)^2}$

2. Clear the denominator by multiplying both sides by $(x - 4)(x^2 + 3)^2$:

   $7x^4 + 44x^2 - 8x + 63 = A(x^2 + 3)^2 + (Bx + C)(x - 4)(x^2 + 3) + (Dx + E)(x - 4)$

3. Expand the terms on the right-hand side:

   We now expand each of these expressions. This will give us a polynomial in $x$, and by comparing coefficients of powers of $x$, we will solve for $A$, $B$, $C$, $D$, and $E$.

   I'll calculate this next.

Part b) Integral Evaluation

We are asked to evaluate the following integral:

$\int \frac{7x^4 + 44x^2 - 8x + 63}{(x - 4)(x^2 + 3)^2} \, dx$

Once we have the partial fraction decomposition from Part a, we can integrate each of the simpler fractions separately. Each term will either be a rational function, which can be integrated using standard techniques (such as substitution or recognizing logarithmic forms).

I'll first complete Part a (finding the partial fraction decomposition), and then proceed to Part b. Let me work through this.The partial fraction decomposition of the given rational function is:

$\frac{7x^4 + 44x^2 - 8x + 63}{(x - 4)(x^2 + 3)^2} = \frac{7}{x - 4} + \frac{2x}{x^2 + 3}$

Part b) Evaluating the Integral

We now need to evaluate the integral:

$\int \frac{7}{x - 4} + \frac{2x}{x^2 + 3} \, dx$

This can be broken into two separate integrals:

1. $\int \frac{7}{x - 4} \, dx$
2. $\int \frac{2x}{x^2 + 3} \, dx$

Let's evaluate these:

1. $\int \frac{7}{x - 4} \, dx = 7 \ln|x - 4|$

2. For $\int \frac{2x}{x^2 + 3} \, dx$, use the substitution $u = x^2 + 3$, leading to:

   $\int \frac{2x}{x^2 + 3} \, dx = \ln|x^2 + 3|$

Thus, the final integral is:

$7 \ln|x - 4| + \ln|x^2 + 3| + C$

where $C$ is the constant of integration.

Would you like any further clarification or additional details?

Follow-up Questions:

1. What are the steps to compare coefficients during partial fraction decomposition?
2. How do you handle improper fractions in partial fraction decomposition?
3. Why do we use substitution in integrals like $\int \frac{2x}{x^2 + 3} \, dx$?
4. Can logarithmic integration results be simplified using properties of logarithms?
5. What are some common techniques for integrating rational functions?

Tip: When evaluating integrals, always look for algebraic manipulations or substitutions that simplify the expression.