a^(2+6) - a^2b^6
= a^2a^6-a^2b^6
=a^2(a^6-b^6), here color(red)(a^2) is common between the terms
=a^2(a^(3xx2)-b^(3xx2))
=a^2[(a^3)^2-(b^3)^2]
color(red)("This is of the form " x^2-y^2 = (x-y)(x+y), "where " x=a^3 " and " y = b^3)
=a^2(a^3-b^3)(a^3+b^3)
Now we factorize (a^3-b^3) and (a^3+b^3):
color(red)("We know that " (a^3-b^3)=(a-b)(a^2+ab+b^2)
color(red)("And "(a^3+b^3)=(a+b)(a^2-ab+b^2)
Then,
a^2(a^3-b^3)(a^3+b^3)=color(blue)(a^2(a-b)(a^2+ab+b^2)(a+b)(a^2-ab+b^2)