Which One Gets the Minus

When the two numbers have opposite signs, adding them works like subtracting. The sign of the sum is the sign of the number that is larger in size. To factor $x^{2}-2x-15$, you need a product of -15 and a sum of -2. The factor pair 5 and 3 has a difference of 2. The sum must be negative, so the larger number, 5, gets the minus sign. The numbers are -5 and 3.

Check: $-5\cdot 3 = -15$ and $-5+3 = -2$. So $x^{2}-2x-15 = \left(x-5\right)\left(x+3\right)$.

When $c$ is negative, ignore the signs at first. Find a factor pair of the size of $c$ whose difference is the size of $b$. If $b$ is negative, put the minus sign on the larger number. If $b$ is positive, put the minus sign on the smaller number. Then check that the two numbers add to $b$ and multiply to $c$.