To solve these quadratic equations, we can use the quadratic formula:

For the first equation: 4x^2 - 5x + 6 = 0
a = 4, b = -5, c = 6

x = (-(-5) ± √((-5)^2 - 446)) / 2*4
x = (5 ± √(25 - 96)) / 8
x = (5 ± √(-71)) / 8

Since the square root of a negative number is not a real number, the first equation does not have real solutions.

For the second equation: 3x^2 - 2x + 3 = 0
a = 3, b = -2, c = 3

x = (-(-2) ± √((-2)^2 - 433)) / 2*3
x = (2 ± √(4 - 36)) / 6
x = (2 ± √(-32)) / 6
x = (2 ± 4√(-2)) / 6
x = (2 ± 4i√2) / 6
x = (1 ± 2i√2) / 3

Therefore, the solutions for the second equation are x = (1 + 2i√2) / 3 and x = (1 - 2i√2) / 3 where i is the imaginary unit.