Question 66105
Let the first integer be n, the next consecutive integer is n+1.
From the problem description, you can write:
{{{n(n+1) - (n + (n+1)) = 181}}} Simplify and solve for n.
{{{n^2+n-2n-1 = 181}}}
{{{n^2-n-182 = 0}}} Solve this quadratic equation by factoring:
{{{(n+13)(n-14) = 0}}} Apply the zero product principle:
{{{n+13 = 0}}} and/or {{{n-14 = 0}}}
If {{{n+13 = 0}}} then {{{n = -13}}} and {{{n+1 = -12}}}
If {{{n-14 = 0}}} then {{{n = 14}}} and n+1 = 15

So you really get two pairs of integers which satisfy the given constraints and since the problem did not restrict the solution to positive integers only, you will have two answers:

14 and 15 is one pair.
-13 and -12 is the other pair.

Check:
{{{(14)(15)-(14+15) = 210 - 29}}} = {{{181}}} OK
{{{(-13)(-12) - (-13 + (-12)) = 156 - (-25)}}} = {{{156 + 25 = 181}}} OK