Answer
The value is, $10x+5h-6$.
Work Step by Step
The function is given below,
$f\left( x \right)=5{{x}^{2}}-6x+1$ (I)
Substitute $x=x+h$ into equation (I):
$\begin{align}
& f\left( x+h \right)=5{{\left( x+h \right)}^{2}}-6\left( x+h \right)+1 \\
& =5\left( {{x}^{2}}+2xh+{{h}^{2}} \right)-6x-6h+1 \\
& =5{{x}^{2}}+10xh+5{{h}^{2}}-6x-6h+1
\end{align}$
Then, calculate
$\frac{f\left( x+h \right)-f\left( x \right)}{h}$ (II)
Substitute the value of $f\left( x+h \right)$ into equation (II).
So,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{5{{x}^{2}}+10xh+5{{h}^{2}}-6x-6h+1-5{{x}^{2}}+6x-1}{h} \\
& =\frac{10xh+5{{h}^{2}}-6h}{h} \\
& =\frac{h\left( 10x+5h-6 \right)}{h}
\end{align}$
Finally,
$\frac{f\left( x+h \right)-f\left( x \right)}{h}=10x+5h-6$ , $h\ne 0$
Hence,
$\frac{f\left( x+h \right)-f\left( x \right)}{h}=10x+5h-6$ where $h\ne 0$.
