Given \( f(x)=x^{2}-4 x \), find \( \frac{f(x+h)-f(x)}{h} \) and simplify. Write all steps on your submission paper for full credit.

To find \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x) = x^2 - 4x \), we will follow these steps: 1. **Calculate \( f(x+h) \)**: \[ f(x+h) = (x+h)^2 - 4(x+h) \] Expanding \( (x+h)^2 \): \[ (x+h)^2 = x^2 + 2xh + h^2 \] Now substituting this into the expression for \( f(x+h) \): \[ f(x+h) = (x^2 + 2xh + h^2) - 4(x + h) \] Expanding \( -4(x + h) \): \[ -4(x + h) = -4x - 4h \] Combining these results: \[ f(x+h) = x^2 + 2xh + h^2 - 4x - 4h \] Rearranging gives: \[ f(x+h) = x^2 - 4x + 2xh + h^2 - 4h \] 2. **Calculate \( f(x+h) - f(x) \)**: We already have \( f(x) = x^2 - 4x \). Now we subtract \( f(x) \) from \( f(x+h) \): \[ f(x+h) - f(x) = (x^2 - 4x + 2xh + h^2 - 4h) - (x^2 - 4x) \] Simplifying this: \[ f(x+h) - f(x) = 2xh + h^2 - 4h \] 3. **Divide by \( h \)**: Now we compute \( \frac{f(x+h) - f(x)}{h} \): \[ \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 4h}{h} \] We can factor \( h \) out of the numerator: \[ = \frac{h(2x + h - 4)}{h} \] Assuming \( h \neq 0 \), we can cancel \( h \): \[ = 2x + h - 4 \] 4. **Final Result**: Thus, the simplified expression for \( \frac{f(x+h) - f(x)}{h} \) is: \[ \boxed{2x + h - 4} \]