Find the product of three consecutive odd integers, if one of them is $(2 m + 1)$ .

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Solution

Formula:
$(x + a) (x + b) (x + c) = x^{3} + (a + b + c) x^{2} + (a b + b c + c a) x + a b c$

Three consecutive odd integers, if one of them is $(2 m + 1)$ are

$(2 m + 1), (2 m + 3),$ and $(2 m + 5)$

Product $= (2 m + 1) (2 m + 3) (2 m + 5)$

$= (2 m)^{3} + (1 + 3 + 5) (2 m)^{2} + (3 + 15 + 5) (2 m) + 15$

$= 8 m^{3} + 36 m^{2} + 52 m + 15$ (Ans)

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