5x^2 + 12x + 3 = 0 would best be solved by using the quadratic formula. This is because it does not factor nicely and the lead coefficient makes it hard to use the square roots method or the completing the square method.
x^2 - 4x = 8 would best be solved with the completing the square method. This is because the first step in completing the square is to put the constant on the opposite side, which is already done for you in this one.
4x^2 - 25 = 0 would best be solved with the square root method. This is because there are only two terms, they are separated by a subtraction symbol, and both are perfect squares.
x^2 - 5x + 6 would best be solved by factoring. This is the case because it is easy to find factors of the constant that add up to the x coefficient (which is what you need to factor). In this case -2 and -3 multiply to 6 and add up to -5.