what are the similarities and differences found in long division of polynomials and synthetic division?

Answer 1

Methods: Factoring, completing the square, quadratic formula.
for these equations:

1. x^2 - x = 6
2. -2x^2 + 5x - 3 = 0
3. 3x^2 + x + 1 = 0
4. 2x(x - 5) = -12

is there any reason to chose one method over the other to solve the equations?