Question 103390
{{{x^2+(x+1)^2+(x+2)^2=77}}}
{{{x^2+(x^2+2x+1)+(x^2+4x+4)=77}}}
{{{3x^2+6x+5=77}}}
{{{3x^2+6x-72=0}}}
{{{x^2+2x-24=0}}}
{{{x+6)(x-4)=0}}}
x=-6 or x=4
Your three consecutive integers are -6,-5,-4 or 4,5,6. 
Let's verify.
{{{4^2+5^2+6^2=77}}}
{{{16+25+36=77}}}
{{{77=77}}}
Good answer.