Consider the quadratic equation y^2-y=6. what is the error with the solution below?

I can not see any "solution below", I can offer you the solutions of the equation you gave

y^2-y-6=0 => y^2-3y+2y-6=0 =>  y(y-3)+ 2(y-3)=0 => (y-3) (y+2)=0

y-3=0 or y+2=0 => y=3 or y= -2

Check 1:  3^2 - 3 = 6 => 9-3=6 => 6=6 Identity prove solution
Check 2:  (-2)^2 - (-2) =6 => 4+2=6 => 6=6 Identity prove solution

Good luck

Here are the answers to the ones that i could see.