We can expand both sides of the equation as:
#(a * a) + (a * 2b) + (5b * a) + (5b * 2b) >= (9b * a) + (9b * b)#
#a^2 + 2ab + 5ab + 10b^2 >= 9ab + 9b^2#
#a^2 + 7ab + 10b^2 >= 9ab + 9b^2#
We can now subtract #color(red)(9ab)# and #color(blue)(9b^2)# from each side of the equation to put all of the variables on one side of the equation while keeping the equation balanced:
#a^2 + 7ab - color(red)(9ab) + 10b^2 - color(blue)(9b^2) >= 9ab - color(red)(9ab) + 9b^2 - color(blue)(9b^2)#
#a^2 + (7 - color(red)(9))ab + (10 - color(blue)(9))b^2 >= 0 + 0#
#a^2 + (-2)ab + 1b^2 >= 0 + 0#
#a^2 - 2ab + b^2 >= 0#
We can now factor this as:
#(a - b)(a - b) >= 0#
#(a - b)^2 >= 0#
Any number squared, by definition, will be a greater than or equal to zero.
If #(a - b) < 0# - then a minus times a minus is a plus and greater than zero.
If #(a - b) > 0# - then a plus times a plus is a plus and greater than zero.
If #(a - b) = 0# - then a zero times zero is zero and equal to zero.
