彻底因式分解下列代数式。

1. Rewrite each squared term as a square of a bracket.

$4(x-y)^2-9(x+y)^2=[2(x-y)]^2-[3(x+y)]^2$

2. Apply the difference of two squares formula and simplify the brackets.

Using $A^2-B^2=(A-B)(A+B)$,
$[2(x-y)]^2-[3(x+y)]^2$
$=[2(x-y)-3(x+y)][2(x-y)+3(x+y)]$
$=(2x-2y-3x-3y)(2x-2y+3x+3y)$
$=(-x-5y)(5x+y)$
$=-(x+5y)(5x+y)$

1. Group the terms and factorise each pair.

$10ax+15ay-8bx-12by=5a(2x+3y)-4b(2x+3y)$

2. Extract the common binomial factor.

$5a(2x+3y)-4b(2x+3y)=(5a-4b)(2x+3y)$