We need to simplify the rational expression $\frac{12x^2 - 19x + 4}{6x^2 - 17x + 12}$ by factoring the numerator and the denominator and then cancelling any common factors.

1. Problem Description

We need to simplify the rational expression $\frac{12x^2 - 19x + 4}{6x^2 - 17x + 12}$ by factoring the numerator and the denominator and then cancelling any common factors.

2. Solution Steps

First, we factor the numerator $12x^2 - 19x + 4$. We look for two numbers that multiply to $12 \times 4 = 48$ and add to $-19$. These two numbers are $-3$ and $-16$. So we rewrite the middle term as $-3x - 16x$.

$12x^2 - 19x + 4 = 12x^2 - 3x - 16x + 4$

Now we factor by grouping:

$12x^2 - 3x - 16x + 4 = 3x(4x - 1) - 4(4x - 1) = (3x - 4)(4x - 1)$

Next, we factor the denominator $6x^2 - 17x + 12$. We look for two numbers that multiply to $6 \times 12 = 72$ and add to $-17$. These two numbers are $-8$ and $-9$. So we rewrite the middle term as $-8x - 9x$.

$6x^2 - 17x + 12 = 6x^2 - 9x - 8x + 12$

Now we factor by grouping:

$6x^2 - 9x - 8x + 12 = 3x(2x - 3) - 4(2x - 3) = (3x - 4)(2x - 3)$

Therefore, we have:

$\frac{12x^2 - 19x + 4}{6x^2 - 17x + 12} = \frac{(3x - 4)(4x - 1)}{(3x - 4)(2x - 3)}$

We can now cancel the common factor $(3x - 4)$:

$\frac{(3x - 4)(4x - 1)}{(3x - 4)(2x - 3)} = \frac{4x - 1}{2x - 3}$

3. Final Answer

$\frac{4x - 1}{2x - 3}$