To factor the quadratic expression \(2x^2 - 11x + 9\), we can use the method of factoring by grouping or the quadratic formula. First, let's look for two numbers that multiply to \(2 \cdot 9 = 18\) and add up to \(-11\).
The numbers that satisfy these conditions are \(-9\) and \(-2\), since:
\[ -9 \times -2 = 18 \quad \text{and} \quad -9 + (-2) = -11. \]
Now we can rewrite the middle term \(-11x\) using these two numbers:
\[ 2x^2 - 9x - 2x + 9. \]
Next, we can group the terms:
\[ (2x^2 - 9x) + (-2x + 9). \]
Now, factor out the common factors from each group:
\[ x(2x - 9) - 1(2x - 9). \]
Notice that \((2x - 9)\) is a common factor:
\[ (2x - 9)(x - 1). \]
So the completely factored form of the expression \(2x^2 - 11x + 9\) is:
\[ \boxed{(2x - 9)(x - 1)}. \]