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

1. Problem Description

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

2. Solution Steps

We want to factor $3x^2 - 11x + 6$ in the form $(ax + b)(cx + d)$ where $a$, $b$, $c$, and $d$ are integers.

We are looking for two numbers that multiply to $3 \cdot 6 = 18$ and add up to $-11$. These two numbers are $-2$ and $-9$.

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 $(x - 3)$:

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

Therefore, $3x^2 - 11x + 6 = (3x - 2)(x - 3)$.

3. Final Answer

$(3x - 2)(x - 3)$