First I will assume that this problem can be reworded as follows: "Two negative integers are five units apart on the number line."
Let the first (the smaller) negative integer be x and the second y. Note that y, being bigger than x, can be represented by y=x+5.
First find the 2 integers. Their product is 126, so x(x+5)=126.
Expanding, x^2 + 5x - 126 = 0. Using the quadratic formula to solve this,
-5 plus or minus sqrt(529) -5 plus or minus 23
x = --------------------------------------- = -----------------------------
2 2
-5 -23
We are interested only in negative integers, so take x = ------------
2
Then x = -28/2, or x = -14. Then y is x+5, or -14+5 = -9.
Check: does the product of -9 and -14 equal 126, as it must? YES.