As given by the question
There are given that the function:
[tex]f(x)=x^2+3[/tex]
Now,
From the given formula:
[tex]\frac{f(x+h)-f(x)}{h}[/tex]
Then,
First find the equation for f(x + h)
So,
[tex]\begin{gathered} f(x)=x^2+3 \\ f(x+h)=(x+h^{})^2+3 \\ f(x+h)=x^2+h^2+2xh+3 \end{gathered}[/tex]
Then,
Put both values into the given formula:
So,
[tex]\frac{f(x+h)-f(x)}{h}=\frac{x^2+h^2+2xh+3-x^2-3}{h}[/tex]
Then,
[tex]\begin{gathered} \frac{f(x+h)-f(x)}{h}=\frac{h^2+2xh}{h} \\ =\frac{h(h^{}+2x)}{h} \\ =h+2x \\ =2x+h \end{gathered}[/tex]
Hence, the difference quotient is 2x + h.
