Question 79539
Let n and n+1 be the two consecutive integers.  We know that 
{{{n(n+1)=n+(n+1)+71}}}
Multiply out the left hand side and simplify the right hand side.

{{{n^2+n=2n+72}}}.  

Now set the equation to 0.

{{{n^2-n-72=0}}}

Factor the left hand side.

{{{(n-9)(n+8)=0}}}

So n=9,-8.

So we have two solutions:
1.  n=9, n+1=10   solution pair (9,10)
2.  n=-8, n+1=-7   solution pair (-8,-7)

Check your answer with both of these to convince yourself that you are right.