## Precalculus (6th Edition) Blitzer

The difference quotient for the provided function is $-6x-3h+1$.
Consider the provided function: $f\left( x \right)=-3{{x}^{2}}+x-1$. Now, substitute $x=x+h$ in the above equation to find $f\left( x+h \right)$ That is, \begin{align} & f\left( x+h \right)=-3{{\left( x+h \right)}^{2}}+\left( x+h \right)-1 \\ & =-3{{x}^{2}}-6xh-3{{h}^{2}}+x+h-1 \end{align} Now, apply the difference quotient formula, \begin{align} & \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{-3{{x}^{2}}-6xh-3{{h}^{2}}+x+h-1-\left( -3{{x}^{2}}+x-1 \right)}{h} \\ & =\frac{-2{{x}^{2}}-4xh-2{{h}^{2}}-x-h+3+2{{x}^{2}}+x-3}{h} \\ & =\frac{-6xh-3{{h}^{2}}+h}{h} \\ & =\frac{h\left( -6x-3h+1 \right)}{h} \end{align} Further solve and get, $\frac{f\left( x+h \right)-f\left( x \right)}{h}=-6x-3h+1$ Hence, the difference quotient for the provided function is $-6x-3h+1$.