Lauren B. answered • 10/02/19

Enthusiastic Physics, Chemistry and Math Tutor with 10yrs experience

Average rate of change is the slope of the line between points

(f(x+h)-f(x))/(x+h-x)<-(denominator simplifies to just h)

plug "x+h" and "x" into f(x)

(4(x+h)^{2}-9-(4x^{2}-9))/(h)

Expand:

(4(x^{2}+2xh+h^{2})-9-(4x^{2}-9))/(h)

Distribute

(4x^{2}+8xh+4h^{2}-9-4x^{2}+9))/(h)

Combine like terms

(8xh+4h^{2}))/(h)

Factor an h our of the numerator

h(8x+4h)/h

Cancel h

8x+4h