Question 1001799
You can try  n, n+2, n+4 as the "odd" integers and see what happens.
{{{n(n+2)(n+4)=-10+n^3}}}.  Solve for n.  Does this work?


{{{n(n^2+6n+8)=n^3-10}}}
{{{n^3+6n^2+8n=n^3-10}}}
{{{6n^2+8n=-10}}}
{{{3n^2+4n+5=0}}}
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(3n  1)(n  5) NO
(3n  5)(n  1) NO
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Use formula for quadratic general solution
{{{n=(-4+- sqrt(16-4*3*5))/6}}}---NO because discriminant is negative.



Try ensuring that the numbers are ODD.
They should be like 2n+1, 2n+3, 2n+5.
The description:
{{{(2n+1)(2n+3)(2n+5)=(2n+1)^3-10}}}
{{{8n^3+36n^2+46n+15=8n^3+12n^2+6n-9}}}
{{{36n^2+46n+15=12n^2+6n-9}}}
A few more simplifications steps,
{{{3n^2+5n+3=0}}}
NOW, plug values into general solution formula for  a quadratic equation and begin simplifying:
{{{n=(-5+- sqrt(25-36))/6}}}
Notice that the discriminant IS NEGATIVE.
<b>NO SOLUTION</b>