To solve this equation, we need to first simplify the left side of the equation.
Given equation:
3x² + 1 = 2x * (x + 1)

Expanding the expression on the right side:
3x² + 1 = 2x² + 2x

Now, bring all terms to one side of the equation:
3x² - 2x² + 2x - 1 = 0
x² + 2x - 1 = 0

Now, we have a quadratic equation in the form of ax² + bx + c = 0.
To solve this, we can use the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a

In our case:
a = 1, b = 2, c = -1

Plugging in the values:
x = [-2 ± √(2² - 41(-1))] / 2*1
x = [-2 ± √(4 + 4)] / 2
x = [-2 ± √8] / 2
x = (-2 ± 2√2) / 2
x = -1 ± √2

So, the solutions to the equation are:
x = -1 + √2 and x = -1 - √2