We are asked to perform the addition and simplify the expression:

$\frac{3}{4x^2 + 4x + 1} + \frac{5}{2x^2 - x - 1}$

Step 1: Find a common denominator

To add these fractions, we need a common denominator. The denominators are $4x^2 + 4x + 1$ and $2x^2 - x - 1$. The common denominator will be the product of these two polynomials:

$(4x^2 + 4x + 1)(2x^2 - x - 1)$

Step 2: Cross-multiply and rewrite the numerators

To express the fractions with the common denominator, we multiply each numerator by the other fraction’s denominator:

$\frac{3(2x^2 - x - 1)}{(4x^2 + 4x + 1)(2x^2 - x - 1)} + \frac{5(4x^2 + 4x + 1)}{(4x^2 + 4x + 1)(2x^2 - x - 1)}$

Step 3: Simplify the numerators

First, expand the numerators:

1. Expand $3(2x^2 - x - 1)$: $3(2x^2 - x - 1) = 6x^2 - 3x - 3$

2. Expand $5(4x^2 + 4x + 1)$: $5(4x^2 + 4x + 1) = 20x^2 + 20x + 5$

Step 4: Combine the numerators

Now, combine the expanded numerators:

$(6x^2 - 3x - 3) + (20x^2 + 20x + 5) = 26x^2 + 17x + 2$

Step 5: Final expression

The resulting fraction is:

$\frac{26x^2 + 17x + 2}{(4x^2 + 4x + 1)(2x^2 - x - 1)}$

This is the simplified form of the addition. Further factorization of the denominator or numerator may not be possible unless requested.

Would you like more details on this solution?

Here are five related questions to explore:

1. How do you find the least common denominator when adding rational expressions?
2. Can the denominator be further factored in this problem?
3. How can you check if two polynomials can be simplified or factored?
4. What are the common methods for simplifying complex fractions?
5. How do you determine if a rational expression is in its simplest form?

Tip: Always look for common factors in both the numerator and denominator before multiplying polynomials to simplify the process.