This is just one way to solve this problem.
First, let's combine everything to one fraction, like this:
#((4x^2 - y^2)(4x^2 - 9xy - 3y^2))/((8x^2 + 10xy + 3y^2)(2x^2 - 5xy - 3y^2))#
Now, let's focus just on the numerator and distribute it.
#4x^2 * 4x^2 = 16x^4#
#4x^2 * -9xy = -36x^3y#
#4x^2 * -3y^2 = -12x^2y^2#
#-y^2 * 4x^2 = -4x^2y^2#
#-y ^2 * -9xy = 9xy^3#
#-y^2 * -3y^2 = 3y^4#
When we put all of these together, we get #16x^4-36x^3y-12x^2y^2-4x^2y^2+9xy^3+3y^4#
The only things in common here are #-12x^2y^2# and #-4x^2y^2#, so #-12x^2y^2 -4x^2y^2 = -16x^2y^2#
So the numerator of the fraction is #16x^4 - 36x^3y - 16x^2y^2 + 9xy^3 + 3y^4#
Now let's look at the denominator #(8x^2 + 10xy + 3y^2)(2x^2 - 5xy - 3y^2)# and distribute it:
#8x^2 * 2x^2 = 16x^4#
#8x^2 * -5xy = -40x^3y#
#8x^2 * -3y^2 = -24x^2y^2#
#10xy * 2x^2 = 20x^3y#
#10xy * -5xy = -50x^2y^2#
#10xy * -3y^2 = -30xy^3#
#3y^2 * 2x^2 = 6x^2y^2 #
#3y^2 * -5xy = -15xy^2#
#3y^2 * -3y^2 = -9y^4#
When we put all of these together, we get #16x^4 - 40x^3y - 24x^2y^2 + 20x^3y - 50x^2y^2 - 30xy^3 + 6x^2y^2 - 15xy^2 - 9y^4#
#-24x^2y^2, -50x^2y^2, and 6x^2y^2# are all in common, so:
#-24x^2y^2 -50x^2y^2 + 6x^2y^2 = -68x^2y^2#
Also, #-40x^3y# and #20x^3y# are in common, so:
#-40x^3y + 20x^3y = -20x^3y#
So now our denominator looks like this:
#16x^4 - 20x^3y - 68x^2y^2 - 30xy^3 - 15xy^2 - 9y^4#
When we put the numerator and denominator together, it looks like this:
#(16x^4 - 36x^3y - 16x^2y^2 + 9xy^3 + 3y^4)/(16x^4 - 20x^3y - 68x^2y^2 - 30xy^3 - 15xy^2 - 9y^4)#
Hope this helps and that I didn't make any mistakes!
