To find the least common denominator (LCD) of the two rational expressions

$\frac{8x}{3x^2 - 11x + 6} \quad \text{and} \quad \frac{3}{3x^2 + 10x - 8},$

we need to factor the denominators of each expression:

1. First denominator: $3x^2 - 11x + 6$

   We look for two numbers that multiply to $3 \cdot 6 = 18$ and add up to $-11$. These numbers are $-9$ and $-2$. Thus, we can factor as:

   $3x^2 - 11x + 6 = (3x + 1)(x - 6)$

2. Second denominator: $3x^2 + 10x - 8$

   We look for two numbers that multiply to $3 \cdot (-8) = -24$ and add up to $10$. These numbers are $12$ and $-2$. Thus, we factor as:

   $3x^2 + 10x - 8 = (3x - 2)(x + 4)$

The LCD is the product of all distinct linear factors from both factored denominators:

$\text{LCD} = (3x + 1)(x - 6)(3x - 2)(x + 4)$

This is the least common denominator for the given rational expressions.

Let me know if you'd like further details or have any questions!

Here are five related questions to deepen your understanding:

1. How do you factor quadratic expressions in general?
2. How would you add or subtract rational expressions with different denominators?
3. What happens when one of the denominators is already a factor of the other?
4. How can you simplify the final expression after finding the LCD?
5. How do you solve equations involving rational expressions?

Tip: Always check if the quadratic can be factored by looking for common factors before applying more advanced factoring methods!