3(х-2)(х+4)=2х+х

To solve the equation \( 3(x-2)(x+4) = 2x + x \), we first simplify the right side of the equation:

\[ 2x + x = 3x \]

Now we can rewrite the equation:

\[ 3(x-2)(x+4) = 3x \]

Next, we can divide both sides of the equation by 3 (assuming \( x \neq 0 \)):

\[ (x-2)(x+4) = x \]

Now, let's expand and simplify the left side:

\[ x^2 + 4x - 2x - 8 = x \]

This simplifies to:

\[ x^2 + 2x - 8 = x \]

Next, we will move all terms to one side of the equation:

\[ x^2 + 2x - x - 8 = 0 \]

This simplifies to:

\[ x^2 + x - 8 = 0 \]

Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -8 \):

1. Calculate the discriminant:

\[ b^2 - 4ac = 1^2 - 4(1)(-8) = 1 + 32 = 33 \]

2. Now, apply the quadratic formula:

\[ x = \frac{-1 \pm \sqrt{33}}{2(1)} = \frac{-1 \pm \sqrt{33}}{2} \]

Thus, the solutions for \( x \) are:

\[ x = \frac{-1 + \sqrt{33}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{33}}{2} \]

These are the solutions to the equation \( 3(x-2)(x+4) = 2x + x \).