Find the difference quotient of \( f \), that is, find \( \frac{f(x+h)-f(x)}{h}, h \neq 0 \), for the following function. Be sure to simplify. \( f(x)=-2 x^{2}+3 x-3 \)

To find the difference quotient \( \frac{f(x+h) - f(x)}{h} \), we first need to calculate \( f(x+h) \). Given \( f(x) = -2x^2 + 3x - 3 \), we substitute \( x+h \) for \( x \): \[ f(x+h) = -2(x+h)^2 + 3(x+h) - 3 \] Expanding \( (x+h)^2 \): \[ f(x+h) = -2(x^2 + 2xh + h^2) + 3x + 3h - 3 \] \[ = -2x^2 - 4xh - 2h^2 + 3x + 3h - 3 \] Now we can substitute for \( f(x) \): \[ f(x+h) - f(x) = (-2x^2 - 4xh - 2h^2 + 3x + 3h - 3) - (-2x^2 + 3x - 3) \] Simplifying this, we get: \[ = -4xh - 2h^2 + 3h \] Now, we divide by \( h \): \[ \frac{-4xh - 2h^2 + 3h}{h} = -4x - 2h + 3 \quad \text{(for } h \neq 0\text{)} \] Thus, the simplified difference quotient is: \[ -4x - 2h + 3 \]