Factorise the following:

x²y² - 8xy - 48

To factorise x²y² - 8xy - 48, we want to find two real numbers whose sum is -8 and product is -48. By trial, we see that (-12) + 4 = -8 and (-12) × 4 = -48.

∴ x²y² - 8xy - 48 = x²y² - 12xy + 4xy - 48

= xy(xy - 12) + 4(xy - 12)

= (xy - 12)(xy + 4).

Hence, x²y² - 8xy - 48 = (xy - 12)(xy + 4).