Difference quotient = 

First calculate t(x+h)

t(x) = x³ + 2

t(x+h) = (x+h)³ + 2 = (x+h)(x+h)(x+h) + 2 = (x+h)(x²+2hx+h²) + 2 =

x³ + 2hx² + h²x + hx² + 2h²x + h² + 2 =

x³ + 3hx² + 3h²x + h³ + 2

Now substitute  for 

and  for 

in the difference quotient expression:


   


Remove the parentheses on top:


  
Cancel the x³ with the -x³ and the +2 with the -2:





Factor h out of the top:



Cancel the h's





That's it!

Edwin