The difference quotient is
[tex]\dfrac{F(x+h)-F(x)}h=\dfrac{(2(x+h)^3-7(x+h)+1)-(2x^3-7x+1)}h[/tex]
Expand the numerator and you'll see cancellation of a factor of [tex]h[/tex] (assuming [tex]h\neq0[/tex]):
[tex]\dfrac{F(x+h)-F(x)}h=\dfrac{2x^3+6x^2h+6xh^2+2h^3-7x-7h+1-2x^3+7x-1}h[/tex]
[tex]\dfrac{F(x+h)-F(x)}h=\dfrac{6x^2h+6xh^2+2h^3-7h}h[/tex]
[tex]\dfrac{F(x+h)-F(x)}h=6x^2-7+6xh+2h^2[/tex]
