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. \( \frac{f(x+h)-f(x)}{h}= \) (i)

To find the difference quotient \( \frac{f(x+h)-f(x)}{h} \), we first need to evaluate \( f(x+h) \). The function \( f(x) \) is given as \( f(x) = -3x^2 + 3x - 7 \). Now, substituting \( x + h \) into the function: \[ f(x+h) = -3(x+h)^2 + 3(x+h) - 7 \] Expanding \( (x+h)^2 \): \[ (x+h)^2 = x^2 + 2xh + h^2 \] Now we can insert this into the function: \[ f(x+h) = -3(x^2 + 2xh + h^2) + 3x + 3h - 7 \] Distributing \( -3 \): \[ f(x+h) = -3x^2 - 6xh - 3h^2 + 3x + 3h - 7 \] Now, we will collect all terms: \[ f(x+h) = -3x^2 + 3x - 7 - 6xh - 3h^2 + 3h \] \[ f(x+h) = (-3x^2 + 3x - 7) - 6xh - 3h^2 + 3h \] \[ f(x+h) = f(x) - 6xh - 3h^2 + 3h \] Now we'll substitute \( f(x) \) back into the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{(-6xh - 3h^2 + 3h)}{h} \] Now we can simplify the expression: \[ \frac{-6xh - 3h^2 + 3h}{h} = -6x - 3h + 3 \] Thus, the simplified difference quotient is: \[ \frac{f(x+h)-f(x)}{h} = -6x + 3 - 3h \] So, the final answer is: \[ \frac{f(x+h)-f(x)}{h} = -6x + 3 - 3h \]