Verified answer

f(x+deltax)-f(x)

if you do this distribution right, you will come up with answer A

and I will apologize because I was one of those who told you it was C, but the difference quotient delta y over delta x is a little different then dy/dx in that it is only instantaneous when delta x goes to zero (consequently causing delta y to go to zero if there is a relationship to x)

∆y is defined as the numerator of the difference quotient

∂y/∂x = lim (∆x --> 0) [ ∆y / ∆x]

∆y = ( f(x + ∆x) - f(x) )

∆y = ( 3x^2 + 6x(∆x) + 3(∆x)^2 + 2x + 2(∆x) - 3x^2 - 2x)

∆y = 6x(∆x) + 3(∆x)^2 + 2(∆x)

y = 3x^2 + 2x

y + ∆y = 3(x + ∆x)^2 + 2(x + ∆x)

y + ∆y - y = 3(x + ∆x)^2 + 2(x + ∆x) - 3x^2 - 2x

∆y = 3(x^ 2 + ∆x^2 + 2x∆x) + 2(x + ∆x) - 3x^2 - 2x

= 3x^ 2 + 3(∆x)^2 + 6x(∆x) + 2x + 2(∆x) - 3x^2 - 2x

= 3(∆x)^2 + 6x(∆x) + 2(∆x)