Let's expand the expression:

(3x - 1)(4x + 3) + 2x(3x - 1) = 0

Expanding the first part using the distributive property:

= 12x^2 + 9x - 4x - 3 + 6x^2 - 2x

Combining like terms:

= 12x^2 + 5x - 3 + 6x^2 - 2x

= 18x^2 + 3x - 3

Now we have:

18x^2 + 3x - 3 = 0

This is a quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the solutions:

x = [-b ± √(b^2 - 4ac)] / 2a

In this case, a = 18, b = 3, and c = -3. Plugging these values into the formula:

x = [-3 ± √(3^2 - 4(18)(-3))] / 2(18)

x = [-3 ± √(9 + 216)] / 36

x = [-3 ± √225] / 36

x = [-3 ± 15] / 36

This gives us two possible solutions:

x = (12/36) = 1/3

x = (-18/36) = -1/2

Therefore, the solutions to the equation are x = 1/3 and x = -1/2.