Question 111627
First integer - N
Next consecutive integer - N+1
{{{N^2+(N+1)^2=11+N}}}
{{{N^2+(N^2+2N+1)=11+N}}}
{{{2N^2+2N+1=11+N}}}
{{{2N^2+N-10=0}}}
{{{(2N+5)(N-2)=0}}}
The solutions are N=-5/2 and N=2
Since you require the solution to be an integer, we only accept the second solution (N=2).
Check the answer.
{{{N^2+(N+1)^2=11+N}}}
{{{2^2+3^2=11+2}}}
{{{4+9=13}}}
{{{13=13}}}
True statement.
Good answer. 
The integers are 2 and 3.