We need to factor the quadratic expression $3x^2 - 11x + 6$.

1. Problem Description

We need to factor the quadratic expression $3x^2 - 11x + 6$.

2. Solution Steps

We are looking for two numbers that multiply to $3 \cdot 6 = 18$ and add up to $-11$.

The two numbers are $-2$ and $-9$ since $(-2) \cdot (-9) = 18$ and $(-2) + (-9) = -11$.

Now, rewrite the middle term using these two numbers:

$3x^2 - 11x + 6 = 3x^2 - 9x - 2x + 6$.

Factor by grouping:

$3x^2 - 9x - 2x + 6 = 3x(x - 3) - 2(x - 3)$.

Factor out the common factor $(x - 3)$:

$3x(x - 3) - 2(x - 3) = (3x - 2)(x - 3)$.

3. Final Answer

The factored form of the quadratic expression $3x^2 - 11x + 6$ is $(3x - 2)(x - 3)$.