Graph the quadratic. If the graphed curve does not cross the "x" axis, or even touch it, the roots are complex. For real world problems this means that the quadratic cannot be factored, because complex roots are not useful for real world problems. If the graph does cross the "x" axis, the point (p) where the curve crosses the "x" axis is a root of the equation and x - p is a possible factor of the quadratic. If the curve just touches the "x" axis at a point, the quadratic has a double root -- two roots but they are both the same. The graph of Y = 4x^2 -4x + 1 just touches the "x" axis at (1/2, 0) so 4x^2 -4x + 1 = (2x -1)^2.
Use the first and last numbers in the quadratic to generate candidate factors. Factors of the first and last numbers in the quadratics will be the first and last numbers in the factors. For example, in 4x^2 - 6x + 2 the first number is 4, which has factors 1, 2 and 4. The last number is 2, which has factors 1 and 2. The candidate factors are x - 1, x + 1, x - 2, x + 2, 2x - 1, 2x + 1, 2x - 2, 2x + 2, 4x - 1, 4x + 1, 4x - 2 and 4x + 2.
Try all of the candidate factors and you will find that 4X^2 - 6X + 2 has factors 2X - 1 and 2X + 2. Guided by the graph of the quadratic, you would try one of these factors first and not have to try them all.