The problem requires us to simplify the given equation $2x(x+3) - 9 = 4x - 8$ and then solve it using the quadratic formula.

1. Problem Description

The problem requires us to simplify the given equation $2x(x+3) - 9 = 4x - 8$ and then solve it using the quadratic formula.

2. Solution Steps

First, we expand and simplify the equation:

$2x(x+3) - 9 = 4x - 8$

$2x^2 + 6x - 9 = 4x - 8$

Subtract $4x$ from both sides:

$2x^2 + 6x - 4x - 9 = -8$

$2x^2 + 2x - 9 = -8$

Add 8 to both sides:

$2x^2 + 2x - 9 + 8 = 0$

$2x^2 + 2x - 1 = 0$

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

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

In our equation $2x^2 + 2x - 1 = 0$, we have $a = 2$, $b = 2$, and $c = -1$. Plugging these values into the quadratic formula:

$x = \frac{-2 \pm \sqrt{2^2 - 4(2)(-1)}}{2(2)}$

$x = \frac{-2 \pm \sqrt{4 + 8}}{4}$

$x = \frac{-2 \pm \sqrt{12}}{4}$

$x = \frac{-2 \pm 2\sqrt{3}}{4}$

Now we simplify by dividing both the numerator and denominator by 2:

$x = \frac{-1 \pm \sqrt{3}}{2}$

So, the two solutions are $x = \frac{-1 + \sqrt{3}}{2}$ and $x = \frac{-1 - \sqrt{3}}{2}$.

3. Final Answer

$x = \frac{-1 + \sqrt{3}}{2}, \frac{-1 - \sqrt{3}}{2}$