Let's go through each part of the problem:

Part (a)

$\frac{5}{10v - 6} - \frac{5}{3v}$

To perform this subtraction, we first find a common denominator for the fractions:

1. Factorize the denominators: $10v - 6 = 2(5v - 3)$ and $3v$.
2. The common denominator will be $6v(5v - 3)$.

Rewrite each fraction with this common denominator:

$\frac{5 \cdot 3v}{6v(5v - 3)} - \frac{5 \cdot 2(5v - 3)}{6v(5v - 3)}$

Simplify and combine the terms in the numerator.

Part (b)

$\frac{6x^2 - 38x + 60}{3x - 10} \div \frac{6x^3 - 18x^2}{3x^3 - 27x^2}$

For division of fractions, multiply by the reciprocal of the second fraction:

$\frac{6x^2 - 38x + 60}{3x - 10} \times \frac{3x^3 - 27x^2}{6x^3 - 18x^2}$

1. Factorize both the numerator and the denominator of each fraction.
2. Simplify where possible and then multiply.

Part (c)

$\frac{11x^5 - 8x^4 + 6x^3 - 12x^2 + 6x}{x^2 + 1}$

For this expression, divide each term in the numerator by $x^2 + 1$ directly, or simplify by polynomial division if possible.

Would you like further detailed steps for each part, or is there a specific part you'd like to focus on?

Follow-up Questions:

1. Would you like a step-by-step solution for simplifying complex fractions?
2. Do you need more explanation on polynomial division?
3. Are you interested in understanding how to factorize polynomials?
4. Would you like guidance on finding common denominators?
5. Are you curious about the applications of these types of expressions?

Tip:

Always factorize completely to simplify complex fractions and make operations like addition, subtraction, multiplication, and division more manageable.