Answer
The required solution is $8x+4h-5$.
Work Step by Step
Replace $ f\left( x \right)$ with $ f\left( x+h \right)$, that is., ${{x}^{2}}={{\left( x+h \right)}^{2}}\text{and }x=x+h $
Then, $\begin{align}
& f\left( x+h \right)=4{{(x+h)}^{2}}-5\left( x+h \right)-2 \\
& f\left( x+h \right)=4\left( {{x}^{2}}+2xh+{{h}^{2}} \right)-5x-5h-2 \\
& f\left( x+h \right)=4{{x}^{2}}+8xh+4{{h}^{2}}-5x-5h-2 \\
\end{align}$
Substitute the values of $ f\left( x+h \right)$ in the provided equation. That is, $\begin{align}
& f\left( x \right)=\frac{f\left( x+h \right)-f\left( x \right)}{h} \\
& f\left( x \right)=\frac{4{{x}^{2}}+8xh+4{{h}^{2}}-5x-5h-2-\left( 4{{x}^{2}}-5x-2 \right)}{h} \\
& f\left( x \right)=\frac{4{{x}^{2}}+8xh+4{{h}^{2}}-5x-5h-2-4{{x}^{2}}+5x+2}{h} \\
& f\left( x \right)=\frac{8xh+4{{h}^{2}}-5h}{h} \\
&
\end{align}$
Solving further to get, $\begin{align}
& f\left( x \right)=\frac{h\left( 8x+4h-5 \right)}{h} \\
& =8x+h-5
\end{align}$
Thus, the simplified form of the provided expression is $8x+4h-5$.
