Question

If f(x)=x^2+3x+4 which of the number choices represents (f(x+h)-f(x))/h?

Answer 1

In case you're not already aware, the expression $\frac{f(x+h)-f(x)}h$ is called the "difference quotient" and represents the average rate of change of a function $f$ over an interval $[x,x+h]$.

For the function $f(x)=x^2+3x+4$, by substituting $x+h$ we get

$f(x+h)=(x+h)^2+3(x+h)+4=(x^2+2xh+h^2)+(3x+3h)+4=x^2+3x+4+(2x+3)h+h^2$

Then the difference quotient is

$\dfrac{f(x+h)-f(x)}h=\dfrac{(x^2+3x+4+(2x+3)h+h^2)-(x^2+3x+4)}h$

$=\dfrac{(2x+3)h+h^2}h=2x+3+h$

where the last equality holds as long as $h\neq0$.