To solve this rational expression, we first analyze the given terms:

The expression is:

$\frac{3x + 2}{3x^2 - x - 2} + \frac{12x - 6}{(x - 3)(2x - 1)}.$

Step 1: Factorize the denominators

For the first denominator $3x^2 - x - 2$:

We factorize $3x^2 - x - 2$ by splitting the middle term: $3x^2 - x - 2 = (3x + 2)(x - 1).$ So, the first term becomes: $\frac{3x + 2}{(3x + 2)(x - 1)} = \frac{1}{x - 1}.$

For the second denominator $(x - 3)(2x - 1)$:

This is already in its factored form.

Step 2: Combine the fractions

Now the expression becomes: $\frac{1}{x - 1} + \frac{12x - 6}{(x - 3)(2x - 1)}.$

The least common denominator (LCD) is: $(x - 1)(x - 3)(2x - 1).$

Step 3: Rewrite both fractions with the LCD

First term:

$\frac{1}{x - 1} = \frac{(x - 3)(2x - 1)}{(x - 1)(x - 3)(2x - 1)}.$

Second term:

$\frac{12x - 6}{(x - 3)(2x - 1)} = \frac{(12x - 6)(x - 1)}{(x - 1)(x - 3)(2x - 1)}.$

Now the expression is: $\frac{(x - 3)(2x - 1) + (12x - 6)(x - 1)}{(x - 1)(x - 3)(2x - 1)}.$

Step 4: Simplify the numerator

Expand both terms in the numerator:

1. Expand $(x - 3)(2x - 1)$: $(x - 3)(2x - 1) = 2x^2 - x - 6x + 3 = 2x^2 - 7x + 3.$

2. Expand $(12x - 6)(x - 1)$: $(12x - 6)(x - 1) = 12x^2 - 12x - 6x + 6 = 12x^2 - 18x + 6.$

Now add the two results: $(2x^2 - 7x + 3) + (12x^2 - 18x + 6) = 14x^2 - 25x + 9.$

Step 5: Final simplified expression

The combined fraction is: $\frac{14x^2 - 25x + 9}{(x - 1)(x - 3)(2x - 1)}.$

Final Answer:

$\frac{14x^2 - 25x + 9}{(x - 1)(x - 3)(2x - 1)}.$

Let me know if you want any further explanation or detailed steps.

Related Questions:

1. How do you factor quadratic expressions efficiently?
2. What are common techniques for finding the least common denominator of rational expressions?
3. How do you verify the correctness of the final simplified fraction?
4. What strategies help simplify large polynomial numerators in rational expressions?
5. How does partial fraction decomposition apply to rational expressions?

Tip:

Always check for opportunities to factorize the numerator and denominator further for a fully reduced form!