To simplify the expression \(\frac{2x^2 - 7x - 4}{x - 4}\), we can start by factoring the numerator, \(2x^2 - 7x - 4\).
First, we'll try to factor the quadratic:
-
We need two numbers that multiply to \(2 \cdot (-4) = -8\) and add up to \(-7\). The numbers \(-8\) and \(1\) work because: \[ -8 \times 1 = -8 \quad \text{and} \quad -8 + 1 = -7 \]
-
Rewrite the quadratic by splitting the middle term: \[ 2x^2 - 8x + x - 4 \]
-
Now, factor by grouping: \[ (2x^2 - 8x) + (x - 4) = 2x(x - 4) + 1(x - 4) \]
-
Factor out the common term \((x - 4)\): \[ (2x + 1)(x - 4) \]
Now, substitute back into the original expression: \[ \frac{2x^2 - 7x - 4}{x - 4} = \frac{(2x + 1)(x - 4)}{x - 4} \]
Assuming \(x \neq 4\) (to avoid division by zero), we can cancel the \((x - 4)\) terms: \[ 2x + 1 \]
Thus, the simplified form of the expression is: \[ 2x + 1 \]
In standard form, the answer is: \[ \boxed{2x + 1} \]