Sure, let's solve each equation one by one.

1. \((x + 2)^2 - x(x + 3) = 2\)

2. \((x + 2)(x - 2) - (x + 1)^2 = 7\)

3. \(6x - (2x + 1)(3x + 2) + 3 = 0\)

4. \((x + 3)(x + 1) - (x - 1)(2x + 1) = 0\)

5. \((x + 1)^2 + (x - 9)(x - 3) - (x - 7)^2 = 0\)

6. \((x - 2)(x + 3)(x + 2)(x - 3) + x^4 = 0\)

7. \((x + 2)(x - 3) - (x^2 - x + 1) = 23\)

Now, we'll solve each equation:

1. \((x + 2)^2 - x(x + 3) = 2\)

Expanding the terms, we have:

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

Simplifying:

\[ x + 4 = 2 \]

So, \( x = -2 \).

2. \((x + 2)(x - 2) - (x + 1)^2 = 7\)

Expand and simplify:

\[ (x^2 - 4) - (x^2 + 2x + 1) = 7 \]

Further simplification gives:

\[ -2x - 5 = 7 \]

\[ -2x = 12 \]

\[ x = -6 \]

3. \(6x - (2x + 1)(3x + 2) + 3 = 0\)

Expanding the terms:

\[ 6x - (6x^2 + 4x + 3x + 2) + 3 = 0 \]

\[ 6x - 6x^2 - 7x - 2 + 3 = 0 \]

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

This is a quadratic equation. We can solve using the quadratic formula:

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

\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-6)(1)}}{2(-6)} \]

\[ x = \frac{1 \pm \sqrt{1 + 24}}{-12} \]

\[ x = \frac{1 \pm \sqrt{25}}{-12} \]

\[ x = \frac{1 \pm 5}{-12} \]

\[ x = \frac{-6}{-12} \quad \text{or} \quad x = \frac{-4}{-12} \]

\[ x = \frac{1}{2} \quad \text{or} \quad x = \frac{1}{3} \]

4. \((x + 3)(x + 1) - (x - 1)(2x + 1) = 0\)

Expanding the terms:

\[ (x^2 + 4x + 3) - (2x^2 + x - 2x - 1) = 0 \]

\[ x^2 + 4x + 3 - 2x^2 - 1 = 0 \]

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

Solving a quadratic equation:

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

\[ x = \frac{-4 \pm \sqrt{16 + 8}}{-2} \]

\[ x = \frac{-4 \pm \sqrt{24}}{-2} \]

\[ x = \frac{-4 \pm 2\sqrt{6}}{-2} \]

\[ x = 2 \pm \sqrt{6} \]

5. \((x + 1)^2 + (x - 9)(x - 3) - (x - 7)^2 = 0\)

Expand each term:

\[ (x^2 + 2x + 1) + (x^2 - 3x - 9x + 27) - (x^2 - 14x + 49) = 0 \]

\[ x^2 + 2x + 1 + x^2 - 12x + 27 - x^2 + 14x - 49 = 0 \]

\[ x^2 + 4x - 21 = 0 \]

Solve this quadratic equation using the quadratic formula:

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

\[ x = \frac{-4 \pm \sqrt{16 + 84}}{2} \]

\[ x = \frac{-4 \pm \sqrt{100}}{2} \]

\[ x = \frac{-4 \pm 10}{2} \]

\[ x = 3 \quad \text{or} \quad x = -7 \]

6. \((x - 2)(x + 3)(x + 2)(x - 3) + x^4 = 0\)

Notice that \((x - 2)(x + 2)\) and \((x + 3)(x - 3)\) are differences of squares:

\[ ((x^2 - 4)(x^2 - 9)) + x^4 = 0 \]

\[ x^4 - 13x^2 + 36 + x^4 = 0 \]

\[ 2x^4 - 13x^2 + 36 = 0 \]

Let \(y = x^2\), then the equation becomes:

\[ 2y^2 - 13y + 36 = 0 \]

Solve this quadratic equation:

\[ y = \frac{13 \pm \sqrt{169 - 288}}{4} \]

\[ y = \frac{13 \pm \sqrt{-119}}{4} \]

Since the discriminant is negative, there are no real solutions.

7. \((x + 2)(x - 3) - (x^2 - x + 1) = 23\)

Expand and simplify:

\[ (x^2 - 3x + 2x - 6) - (x^2 - x + 1) = 23 \]

\[ x^2 - x - 6 - x^2 + x - 1 = 23 \]

\[ -7 = 23 \]

Since this simplifies to a false statement, there is no solution for this equation.

These are the solutions for each given equation.