Question 118980
*[Tex \LARGE \frac{f(x+h)-f(x)}{h} = \frac{{f\left( {x + h } \right) - f\left( x \right)}}{h }]




*[Tex \LARGE \frac{f(x+h)-f(x)}{h} = \frac{(x+h)^2+4(x+h)+5 -\left(x^2+4x+5\right)}{h }] Plug in {{{f(x+h)=(x+h)^2+4(x+h)+5}}} and {{{f(x)=x^2+4x+5}}}



*[Tex \LARGE \frac{f(x+h)-f(x)}{h} = \frac{\left[x^2+2hx+h^2+4x+4h+5\right] -\left(x^2+4x+5\right)}{h }] Foil/Expand {{{(x+h)^2+4(x+h)+5}}}




*[Tex \LARGE \frac{f(x+h)-f(x)}{h} = \frac{2hx+4h+h^2}{h }] Combine like terms (i.e. combine {{{(x^2+4x+5)-(x^2+4x+5)}}} to get 0)


*[Tex \LARGE \frac{f(x+h)-f(x)}{h} = \frac{h\left(2x+4+h\right)}{h }] Factor out an h


*[Tex \LARGE \frac{f(x+h)-f(x)}{h} = \frac{2x+4+h}{1}] Divide {{{h/h}}} to get 1


*[Tex \LARGE \frac{f(x+h)-f(x)}{h} = 2x+4+h] Simplify