How does the sign of the third term relate to the sign of the second term when
factoring? Please show me an example.
When factoring a quadratic f(x) = ax2+bx+c, you are (usually) seeking integers p and q which satisfy
p+q = b
pq = ac
simultaneously. You can check to see whether factoring over the integers is worthwhile by seeing whether the discriminant (b2-4ac) is a perfect square. If so, then try to factor.
Now, if the signs of the coefficients a and c are the same, then the two numbers p and q must both be positive if b is positive, and both be negative if b is negative (why?), whereas if the signs of a and c differ, then p and q must differ in sign. You may be able to find more information if you study certain examples carefully.
Here is a simple one:
f(x) = 2x2-3x-2.
First, the discriminant is in this case (-3)2-4(2)(-2) = 25, which is a square, so factoring over the integers can be pursued. In this case, a = 2 and c = -2 differ in sign, so the integers p and q sought must differ in sign. If we complete the factorization (by whatever method you are used to), we find that p = -4 and q = 1, so that
f(x) = 2x2-3x-2 = 2x2+px+qx-2 = 2x2-4x+x-2 = (2x+1)(x-2).
Observe that the roots of f(x) (-1/2 and 2) also differ in sign. You may want to see if you can state this as a general rule, based on what we just discussed.
I hope this puts you on the path to the solution you seek.