The difference quotient is



First find f(x+h) by substituting (x+h) 
for each x in 

  f(x) = 2x² - 5x + 3 
f(x+h) = 2(x+h)² - 5(x+h) + 3
f(x+h) = 2(x+h)(x+h) - 5(x+h) + 3
f(x+h) = 2(x²+hx+hx+h²) - 5x - 5h + 3
f(x+h) = 2(x²+2hx+h²) - 5x - 5h + 3
f(x+h) = 2x² + 4hx + 2h² - 5x - 5h + 3

No we substitute that for the f(x+h) in



and we have:



and we substitute (2x² - 5x + 3) for f(x), making sure
that we place it inside parentheses, so as not to get
any signs wrong since it is preceded by a - sign:



Now we remove the parentheses by changing the signs inside:



We find that all terms in the numerator which do not
contain the factor h will cancel and we are left with:



That will allow us to factor out an h in the numerator:



Then we can cancel the h's



And we are left with:



Edwin