Find 3 consecutive off integers such that the square of the third integer

Question

increased by twice the product of the first two integers, is the same as 3 more than four times the square of the first integer.

Answer 1

If the integers are x-1, x and x+1,

(x+1)²+2x(x-1)=4(x-1)²+3.

x²+2x+1+2x²-2x=4x²-8x+4+3.

x²-8x+6=0 has no integer solutions: x²-8x+16=10, (x-4)²=10.