p + 1p + 2 = 1 , Find the value of ( p + 2 )^3 + 1 ( p + 2 )^3 - 3 = ?

Answer: Option D

\begin{aligned} & {\text{Give,}} \\ & p + \frac{1}{{p + 2}} = 1 \\ & {\text{Adding 2 both sides}} \\ & p + 2 + \frac{1}{{p + 2}} = 1 + 2 = 3 \\ & {\text{Let}}\left( {p + 2} \right) = a\,\,\&\,\, \frac{1}{{p + 2}} = b \\ & a + b = 3 \\ & {\text{Cubbing both sides}} \\ & {\left( {a + b} \right)^3} = {3^3} \\ & {a^3} + {b^3} + 3ab\left( {a + b} \right) = 27 \\ & {a^3} + {b^3} + 3 \times ab \times 3 = 27 \\ & {a^3} + {b^3} + 9ab = 27......\left( {\text{i}} \right) \\ & {\text{Now,}} \\ & a \times b = \left( {p + 2} \right) \times \frac{1}{{\left( {p + 2} \right)}} \\ & a \times b = 1........\left( {{\text{ii}}} \right) \\ & {\text{Put the a & b value in equation}}\left( {\text{i}} \right) \\ & {a^3} + {b^3} + 9 \times 1 = 27 \\ & {a^3} + {b^3} = 27 - 9 = 18 \\ & \therefore {\left( {p + 2} \right)^3} + \frac{1}{{{{\left( {p + 2} \right)}^3}}} - 3 \\ & = {a^3} + {b^3} - 3 \\ & = 18 - 3 \\ & = 15 \\\end{aligned}