We are asked to solve the equation $3x(x-2) + 4 = 6$ for $x$. This involves expanding the expression, simplifying to a quadratic equation, and then solving for $x$.

1. Problem Description

We are asked to solve the equation $3x(x-2) + 4 = 6$ for $x$. This involves expanding the expression, simplifying to a quadratic equation, and then solving for $x$.

2. Solution Steps

First, expand the left side of the equation:

$3x(x-2) + 4 = 6$

$3x^2 - 6x + 4 = 6$

Next, subtract 6 from both sides of the equation to set it equal to 0:

$3x^2 - 6x + 4 - 6 = 6 - 6$

$3x^2 - 6x - 2 = 0$

Now, we use the quadratic formula to solve for $x$. The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In our equation, $a=3$, $b=-6$, and $c=-2$. Plugging these values into the formula, we get:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(-2)}}{2(3)}$

$x = \frac{6 \pm \sqrt{36 + 24}}{6}$

$x = \frac{6 \pm \sqrt{60}}{6}$

$x = \frac{6 \pm \sqrt{4 \cdot 15}}{6}$

$x = \frac{6 \pm 2\sqrt{15}}{6}$

$x = \frac{3 \pm \sqrt{15}}{3}$

3. Final Answer

$x = \frac{3 + \sqrt{15}}{3}$ or $x = \frac{3 - \sqrt{15}}{3}$