Find the difference quotient \( \frac{f(x+h)-f(x)}{h} \), where \( h \neq 0 \), for the function below. \( f(x)=-3 x^{2}+3 x-7 \) Simplify. your answer as much as possible. \( \frac{f(x+h)-f(x)}{h}= \)

To find the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x)=-3 x^{2}+3 x-7 \), we first need to compute \( f(x+h) \). 1. **Calculate \( f(x+h) \)**: \[ f(x+h) = -3(x+h)^2 + 3(x+h) - 7 \] Expanding this: \[ = -3(x^2 + 2xh + h^2) + 3x + 3h - 7 \] \[ = -3x^2 - 6xh - 3h^2 + 3x + 3h - 7 \] 2. **Subtract \( f(x) \)**: \[ f(x+h) - f(x) = \left(-3x^2 - 6xh - 3h^2 + 3x + 3h - 7\right) - \left(-3x^2 + 3x - 7\right) \] The \( -3x^2 \), \( 3x \), and \( -7 \) terms cancel out: \[ = -6xh - 3h^2 + 3h \] 3. **Factor the difference**: \[ = -3h^2 - 6xh + 3h = h(-6x - 3h + 3) \] 4. **Divide by \( h \)**: Since \( h \neq 0 \), we can divide by \( h \): \[ \frac{f(x+h) - f(x)}{h} = -6x - 3h + 3 \] Thus, the simplified difference quotient is: \[ \frac{f(x+h)-f(x)}{h} = -6x - 3h + 3 \]