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(i)
Expression 1: x + 9
Expression 2: x2 – 9x + 81
Product
= (x + 9)(x2 – 9x + 81)
= (x + 9)(x2 – 9x + 92)
= x3 + 93 [∵ (a + b)(a2 – ab + b2) = a3 + b3]
= x3 + 729
(ii)
Expression 1: 2a – 1
Product = 8a3 – 1
= (2a)3 - 13
= (2a – 1)[(2a)2 + (2a)(1) + (1)2]
[∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= (2a – 1)(4a2 + 2a + 1)
Hence, Expression 2 is (4a2 + 2a + 1)
(iii)
Expression 1: 3 – 5c
Product = 27 – 125c3
= (3)3 – (5c)3
= (3 – 5c)(32 + 3(5c) + (5c)2) [∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= (3 – 5c)(9 + 15c + 25c2)
Hence, Expression 2 is (9 + 15c + 25c2)
(iv)
Expression 1: (a + b + c)
Expression 2: (a + b)2 – (a + b)c + c2
Product
= (a + b + c)[(a + b)2 – (a + b)c + c2]
= [(a + b) + c][(a + b)2 – (a + b)c + c2]
[Using, a3 + b3 = (a + b)(a2 – ab + b2)]
If a = a + b, b = c
= (a + b)3 + c3
(v)
Expression 1: 3x
Expression 2: (2x – 1)2 – (2x – 1) (x + 1) + (x + 1)2
Product
= 3x [(2x – 1)2 – (2x – 1)(x + 1) + (x + 1)2]
As 3x can be written as 2x-1+x+1
= [2x-1+x+1] [(2x – 1)2 – (2x – 1) (x + 1) + (x + 1)2]
[Using, a3 + b3 = (a + b) (a2 – ab + b2)
Here a = 2x-1, b = x+1
= (2x – 1)3 + (x + 1)3
As (a-b)3 =a3 -3a2b+3ab2 – b3
(a + b)3=a3+3a2b+3ab2+b3
= (2x)3 – 1 – 3(2x)2(1) + 3(2x)(1)2 + x3 + 1 + 3x2 + 3x
= 8x3 – 1 – 12x2 + 6x + x3 + 1 + 3x2 + 3x
= 9x3 – 9x2 + 9x
(vi)
Expression 1: 
Expression 2: 
Product
[using, (a + b)3 = (a + b)(a2 – ab + b2)]
(vii)
Expression 1: 4a – 5b
Expression 2: 16a2 + 20ab + 25b2
Product
= (4a – 5b)(16a2 + 20ab + 25b2)
= (4a – 5b)((4a)2 + (4a)(5b) + (5b)2)
[Using, a3 – b3 = (a – b)(a2 + ab + b2)]
= (4a)3 – (5b)3
= 64a3 – 125b3
(viii)
Expression 2: a2b2 + abcd + c2d2
Product
= a3b3 – c3d3
= (ab)3 – (cd)3
= (ab – cd)[(ab)2 + (ab)(cd) + (cd)2]
[Using, a3 – b3 = (a – b)(a2 + ab + b2)]
= (ab – cd)(a2b2 + abcd + c2d2]
Hence, Expression 1 is (ab – cd)
(ix)
Expression 1: 1 – 4y
Product = 1 – 64y3
= 13 – (4y)3
= (1 – 4y)(12 + 1(4y) + (4y)2)
[∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= (1 – 4y)(1 + 4y + 16y2)
Hence, Expression 2 is (1 + 4y + 16y2)
(x)
Expression 1: (2p + 1)
Product = 8(p – 3)3 + 343
= (2(p – 3))3 + 73
[Using, a3 – b3 = (a – b)(a2 + ab + b2)]
= [2(p – 3) + 7][(2(p – 3))2 + 2(p – 3)(7) + 72]
= (2p – 6 + 7)[4(p – 3)2 + 14p – 42 + 49]
= (2p + 1)(4(p2 – 6p + 9) + 14p + 7)
= (2p + 1)(4p2 – 24p + 36 + 14p + 7)
= (2p + 1)(4p2 – 10p + 43)
Hence, Expression 2 is (4p2 – 10p + 43)
(xi)
Expression 1 : m – p
Expression 2: (m + n)2 + (m + n)(n + p) + (n + p)2
Product
= (m – p)[(m + n)2 + (m + n)(n + p) + (n + p)2]
= [m + n – (n + p)]((m + n)2 + (m + n)(n + p) + (n + p)2]
[Using, a3 – b3 = (a – b)(a2 + ab + b2)]
= (m + n)3 – (n + p)3
= (m3 + n3 + 3m2n + 3mn2) – (n3 + p3 + 3n2p + 3np2)
= m3 + n3 + 3m2n + 3mn2 – n3 – p3 – 3n2p – 3np2
= m3 – p3 + 3m2n + 3mn2 – 3n2p – 3np2
(xii)
Expression 1: (3a - 2b)2 + (3a – 2b) × (2a – 3b) + (2a – 3b)2
Expression 2: (a + b)
Product
= (a + b)[(3a - 2b)2 + (3a – 2b) × (2a – 3b) + (2a – 3b)2]
= (3a – 2b – (2a – 3b))[(3a - 2b)2 + (3a – 2b) × (2a – 3b) + (2a – 3b)2]
= (3a – 2b)3 - (2a – 3b)3
[Now,
(a – b)3 = a3 – b3 – 3a2b + 3ab2]
= [(3a)3 – (2b)3 – 3(3a)2(2b) + 3(3a)(2b)2] – [(2a)3 – (3b)3 – 3(2a)2(3b) + 3(2a)(3b)2]
= [27a3 – 8b3 – 54a2b + 36ab2] – [8a3 – 27b3 – 36a2b + 54ab2]
= 27a3 – 8b3 – 54a2b + 36ab2 – 8a3 + 27b3 + 36a2b – 54ab2
= 19a3 + 19b3 – 18a2b – 18b2
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