If x = \(\sqrt {2}\) - 1,

find 3x⁵ + 15x⁴ + 30x³ + 30x² + 15x + 5.

2+ 12\(\sqrt {2}\)

 3x⁵ + 15x⁴ + 30x³ + 30x² + 15x + 5

⇒ 3(x⁵ + 5x⁴ + 10x³ + 10x² + 5x + 1) + 2 ......(i)

We know, (x+1)⁵ = x⁵ + 5x⁴ + 10x³ + 10x² + 5x + 1

eq. (i) becomes 

⇒ 3(x+1)⁵ + 2

⇒ 3(1 + \(\sqrt {2}\) - 1)⁵ + 2

= 3 × \( { \left(\sqrt {2}\right) }^{5} \) + 2

= 2 +12\(\sqrt {2}\)