The greatest whole number, by which the expression n⁴ + 6n³ + 11n² + 6n + 24 is divisible for every natural number n , is -

$$\eqalign{ & {\text{According to the question,}} \cr & {{\text{n}}^4} + 6{{\text{n}}^3} + 11{{\text{n}}^2} + 6{\text{n + 24}} \cr & {\text{put n = 1}} \cr & = 1 + 6 + 11 + 6 + 24 \cr & = 48 \cr & {\text{put n = 2}} \cr & {\text{ = 16 + 48 + 44 + 12 + 24}} \cr & {\text{ = 144}} \cr & {\text{Clearly it is divisible by 48}} \cr} $$