Mathematics
Answer
x3 + 6x2 + 12x + 16 can be written as x3 + 6x2 + 12x + 8 + 8.
Above terms can be written as,
[(x)3 + (3 × 2 × x2) + (3 × 22 × x) + 23] + 8
= [(x)3 + 3 × x × 2(x + 2) + 23] + 23 ……..(i)
We know that,
(a + b)3 = a3 + 3ab(a + b) + b3 ………(ii)
Comparing equation (i) and (ii) we get,
a = x and b = 2.
∴ [(x)3 + 3 × x × 2(x + 2) + 23] + 23 = (x + 2)3 + 23.
We know that,
a3 + b3 = (a + b)(a2 - ab + b2).
∴ (x + 2)3 + 23 = (x + 2 + 2)[(x + 2)2 - (x + 2).2 + 22]
= (x + 4)(x2 + 4 + 4x - 2x - 4 + 4)
= (x + 4)(x2 + 2x + 4)
Hence, x3 + 6x2 + 12x + 16 = (x + 4)(x2 + 2x + 4).