# Prove that (x + 1 + i) (x + 1 – i) (x – 1 – i) (x – 1 – i) = (x4 + 4).

To Prove:

(x + 1 + i) (x + 1 – i) (x – 1 + i) (x – 1 – i) = (x4 + 4)

Taking LHS

(x + 1 + i) (x + 1 – i) (x – 1 + i) (x – 1 – i)

= [(x + 1) + i][(x + 1) – i][(x – 1) + i][(x – 1) – i]

Using (a – b)(a + b) = a2 – b2

= [(x + 1)2 – (i)2] [(x – 1)2 – (i)2]

= [x2 + 1 + 2x – i2](x2 + 1 – 2x – i2]

= [x2 + 1 + 2x – (-1)](x2 + 1 – 2x – (-1)] [ i2 = -1]

= [x2 + 2 + 2x][x2 + 2 – 2x]

Again, using (a – b)(a + b) = a2 – b2

Now, a = x2 + 2 and b = 2x

= [(x2 + 2)2 – (2x)2]

= [x4 + 4 + 2(x2)(2) – 4x2] [(a + b)2 = a2 + b2 + 2ab]

= [x4 + 4 + 4x2 – 4x2]

= x4 + 4

= RHS

LHS = RHS

Hence Proved

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