The Sign Map

Four sign patterns, one map, read c first. Positive c: same signs, and b picks which. x^2 + 7x + 6 takes +1 and +6; x^2 - 5x + 6 takes -2 and -3.

Negative c: opposite signs, hunt by difference, and b's sign lands on the larger number. x^2 + x - 12 takes +4 and -3; x^2 - x - 12 takes -4 and +3.

That is the whole map. Before any hunt, say the cell out loud: same or opposite, and where the minus sits. The pair search then runs through half the candidates with no sign surprises at the end.