The problem requires us to simplify the equation $(x-2)(x+1) = \frac{2x+19}{2}$ and then solve for $x$ using the quadratic formula.

1. Problem Description

The problem requires us to simplify the equation $(x-2)(x+1) = \frac{2x+19}{2}$ and then solve for $x$ using the quadratic formula.

2. Solution Steps

First, expand the left side of the equation:

$(x-2)(x+1) = x^2 + x - 2x - 2 = x^2 - x - 2$

So, we have:

$x^2 - x - 2 = \frac{2x + 19}{2}$

Multiply both sides by 2 to eliminate the fraction:

$2(x^2 - x - 2) = 2x + 19$

$2x^2 - 2x - 4 = 2x + 19$

Move all terms to one side to set the equation to zero:

$2x^2 - 2x - 4 - 2x - 19 = 0$

$2x^2 - 4x - 23 = 0$

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

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

In our quadratic equation $2x^2 - 4x - 23 = 0$, we have $a = 2$, $b = -4$, and $c = -23$.

Plugging these values into the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-23)}}{2(2)}$

$x = \frac{4 \pm \sqrt{16 + 184}}{4}$

$x = \frac{4 \pm \sqrt{200}}{4}$

$x = \frac{4 \pm \sqrt{100 \cdot 2}}{4}$

$x = \frac{4 \pm 10\sqrt{2}}{4}$

Simplify the expression by dividing by 2:

$x = \frac{2 \pm 5\sqrt{2}}{2}$

So the two solutions for $x$ are $x = \frac{2 + 5\sqrt{2}}{2}$ and $x = \frac{2 - 5\sqrt{2}}{2}$.

3. Final Answer

$x = \frac{2 + 5\sqrt{2}}{2}, \frac{2 - 5\sqrt{2}}{2}$