We need to solve the equation:

\[
6x - (2x - 1)(2x + 1) = 2 - (3 + 2x)2
\]

### Step 1: Expand both sides of the equation

#### Left-hand side:
Expand \((2x - 1)(2x + 1)\) using the difference of squares formula:

\[
(2x - 1)(2x + 1) = (2x)^2 - 1^2 = 4x^2 - 1
\]

Now substitute this into the left-hand side:

\[
6x - (4x^2 - 1) = 6x - 4x^2 + 1
\]

#### Right-hand side:
Expand the expression \((3 + 2x)2\):

\[
(3 + 2x)2 = 6 + 4x
\]

So, the right-hand side becomes:

\[
2 - (6 + 4x) = 2 - 6 - 4x = -4 - 4x
\]

### Step 2: Rewrite the equation

The equation is now:

\[
6x - 4x^2 + 1 = -4 - 4x
\]

### Step 3: Move all terms to one side

Move all terms to the left-hand side:

\[
6x - 4x^2 + 1 + 4x + 4 = 0
\]

Simplify the equation:

\[
-4x^2 + 10x + 5 = 0
\]

### Step 4: Solve the quadratic equation

We can solve \(-4x^2 + 10x + 5 = 0\) using the quadratic formula:

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

Here, \(a = -4\), \(b = 10\), and \(c = 5\).

First, calculate the discriminant:

\[
\Delta = b^2 - 4ac = 10^2 - 4(-4)(5) = 100 + 80 = 180
\]

Now, apply the quadratic formula:

\[
x = \frac{-10 \pm \sqrt{180}}{2(-4)} = \frac{-10 \pm \sqrt{180}}{-8}
\]

Simplify \(\sqrt{180}\):

\[
\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}
\]

Substitute this into the formula:

\[
x = \frac{-10 \pm 6\sqrt{5}}{-8}
\]

Now, divide both terms:

\[
x = \frac{10 \mp 6\sqrt{5}}{8}
\]

Simplify:

\[
x = \frac{5 \mp 3\sqrt{5}}{4}
\]

So, the two solutions are:

\[
x = \frac{5 + 3\sqrt{5}}{4} \quad \text{or} \quad x = \frac{5 - 3\sqrt{5}}{4}
\]

Would you like more details on any of these steps?

### Related Questions:
1. What is the discriminant and how does it affect the solutions of a quadratic equation?
2. How can we verify the solutions by substituting them back into the original equation?
3. What are the conditions for a quadratic equation to have real, complex, or repeated solutions?
4. How can we simplify square roots of non-perfect squares like \(\sqrt{180}\)?
5. How can we factor quadratic equations like \(-4x

Learn how to solve the algebraic equation 6x - (2x - 1)(2x + 1) = 2 - (3 + 2x)2 using the difference of squares and the quadratic formula. This step-by-step guide covers essential concepts for solving quadratic equations and is ideal for students in Grades 9-12.

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