Mathematics
Answer
32a2x3 - 8b2x3 - 4a2y3 + b2y3 = 8x3(4a2 - b2) - y3(4a2 - b2)
= (4a2 - b2)(8x3 - y3)
= [(2a)2 - (b)2][(2x)3 - (y)3]
We know that,
a3 - b3 = (a - b)(a2 + ab + b2)
a2 - b2 = (a - b)(a + b).
∴ [(2a)2 - (b)2][(2x)3 - (y)3] = (2a - b)(2a + b)(2x - y)[(2x)2 + 2xy + y2]
= (2a - b)(2a + b)(2x - y)(4x2 + 2xy + y2).
Hence, 32a2x3 - 8b2x3 - 4a2y3 + b2y3 = (2a - b)(2a + b)(2x - y)(4x2 + 2xy + y2).