The problem consists of two parts involving partial fraction decomposition and integration of a rational function. Let's break down each part.

Part (a): Partial Fraction Decomposition

We are asked to decompose the following rational function into simpler fractions: $\frac{3x^4 + 18x^3 + 55x^2 + 78x + 48}{x(x^2 + 3x + 4)^2}$

For this decomposition, the general form would involve writing the numerator as a sum of terms related to the denominator factors:

1. For the linear factor $x$, we use: $\frac{A}{x}$

2. For the repeated quadratic factor $(x^2 + 3x + 4)^2$, we use: $\frac{Bx + C}{x^2 + 3x + 4} \quad \text{and} \quad \frac{Dx + E}{(x^2 + 3x + 4)^2}$

Thus, the decomposition would look like: $\frac{A}{x} + \frac{Bx + C}{x^2 + 3x + 4} + \frac{Dx + E}{(x^2 + 3x + 4)^2}$

Next, to determine the constants $A, B, C, D, E$, we would multiply both sides by the denominator and equate the numerators. This step involves expanding the terms and solving a system of equations based on the powers of $x$.

Part (b): Integral Evaluation

We are asked to evaluate the following integral: $\int \frac{3x^4 + 18x^3 + 55x^2 + 78x + 48}{x(x^2 + 3x + 4)^2} \, dx$

After completing the partial fraction decomposition in part (a), the integral can be broken into simpler parts corresponding to each term in the decomposition. Each term would be integrated individually, typically resulting in logarithmic or arctangent functions for the linear and quadratic terms.

I'll now proceed with solving part (a) by determining the constants for the partial fraction decomposition.The partial fraction decomposition of the given rational function is:

$\frac{3x^4 + 18x^3 + 55x^2 + 78x + 48}{x(x^2 + 3x + 4)^2} = \frac{3}{x} + \frac{4x + 6}{(x^2 + 3x + 4)^2}$

Part (b): Integral Evaluation

Now that we have the partial fraction decomposition, we can break the integral into two parts:

$\int \frac{3}{x} \, dx + \int \frac{4x + 6}{(x^2 + 3x + 4)^2} \, dx$

Let's evaluate these integrals individually:

1. $\int \frac{3}{x} \, dx = 3 \ln|x|$
2. The second integral $\int \frac{4x + 6}{(x^2 + 3x + 4)^2} \, dx$ will likely involve a substitution for the quadratic term.

I'll proceed by evaluating the second integral now.The integral evaluates as follows:

$\int \frac{3x^4 + 18x^3 + 55x^2 + 78x + 48}{x(x^2 + 3x + 4)^2} \, dx = 3 \ln|x| - \frac{2}{x^2 + 3x + 4} + C$

where $C$ is the constant of integration.

Would you like more details on any step, or do you have further questions?

Here are five related questions to expand on this:

1. How would the solution change if the denominator had different factors?
2. What substitution could you use for the second integral involving the quadratic denominator?
3. How does partial fraction decomposition help in solving integrals involving rational functions?
4. What are the conditions where partial fractions apply in integration?
5. How do you handle integrals involving higher-order polynomials in the denominator?

Tip: Always double-check the degree of the numerator compared to the denominator before attempting partial fraction decomposition. If the numerator's degree is greater, perform polynomial long division first.