Answer
The required expression is\[4h-2+8x\]
Work Step by Step
In order to find the value of $\frac{f\left( x+h \right)-f\left( x \right)}{h},$ calculate $f\left( x+h \right)$.
$\begin{align}
& f\left( x+h \right)=4{{\left( x+h \right)}^{2}}-2\left( x+h \right)+7 \\
& =4\left( {{x}^{2}}+{{h}^{2}}+2hx \right)-2x-2h+7 \\
& =4{{x}^{2}}+4{{h}^{2}}+8xh-2x-2h+7
\end{align}$
Now, putting the value $f\left( x+h \right)$ in $\frac{f\left( x+h \right)-f\left( x \right)}{h}$:
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{4{{x}^{2}}+4{{h}^{2}}+8xh-2x-2h+7-\left( 4{{x}^{2}}-2x+7 \right)}{h} \\
& =\frac{4{{x}^{2}}+4{{h}^{2}}+8xh-2x-2h+7-4{{x}^{2}}+2x-7}{h} \\
& =\frac{4{{h}^{2}}-2h+8xh}{h} \\
& =\frac{h\left( 4h-2+8x \right)}{h}
\end{align}$
Hence, the value of $\frac{f\left( x+h \right)-f\left( x \right)}{h}$ is $4h-2+8x$.
