In case you're not already aware, the expression [tex]\frac{f(x+h)-f(x)}h[/tex] is called the "difference quotient" and represents the average rate of change of a function [tex]f[/tex] over an interval [tex][x,x+h][/tex].
For the function [tex]f(x)=x^2+3x+4[/tex], by substituting [tex]x+h[/tex] we get
[tex]f(x+h)=(x+h)^2+3(x+h)+4=(x^2+2xh+h^2)+(3x+3h)+4=x^2+3x+4+(2x+3)h+h^2[/tex]
Then the difference quotient is
[tex]\dfrac{f(x+h)-f(x)}h=\dfrac{(x^2+3x+4+(2x+3)h+h^2)-(x^2+3x+4)}h[/tex]
[tex]=\dfrac{(2x+3)h+h^2}h=2x+3+h[/tex]
where the last equality holds as long as [tex]h\neq0[/tex].
