We have the identities
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
So (5x – 3y)2 – (5x + 3y)2 = (5x)2 – (2 x 5x x 3y) + (3y)2
= 52x2 – 30xy + 32 y2 – [52x2 – 30xy + 32 y2]
= 25x2 – 30xy + 9y2 – [25x2 + 30xy + 9y2]
= 25x2 – 30xy + 9y2 – 25x2 – 30xy – 9y2
= x2 (25 – 25) – xy (30 + 30) + y2 (9 – 9)
= 0x2 – 60xy + 0y2
= – 60 xy
∴ (5x – 3y)2 – (5x + 3y)2 = -60xy