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We are asked to solve for all values of \( x \) in the equation:

\[
\log_{-2x-3}(3x^2 - 18) = 2
\]

### Step-by-Step Solution:

1. **Rewrite the logarithmic equation in exponential form:**

   The logarithmic equation \(\log_b(A) = C\) can be rewritten as:
   \[
   A = b^C
   \]
   Here, \( b = -2x - 3 \), \( A = 3x^2 - 18 \), and \( C = 2 \). Thus:
   \[
   3x^2 - 18 = (-2x - 3)^2
   \]

2. **Simplify both sides:**

   - On the left-hand side:
     \[
     3x^2 - 18 = 3(x^2 - 6)
     \]
   
   - On the right-hand side, expand \( (-2x - 3)^2 \):
     \[
     (-2x - 3)^2 = (2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9
     \]

   Now the equation becomes:
   \[
   3(x^2 - 6) = 4x^2 + 12x + 9
   \]

3. **Expand and rearrange the equation:**

   Expand the left-hand side:
   \[
   3x^2 - 18 = 4x^2 + 12x + 9
   \]

   Bring all terms to one side:
   \[
   3x^2 - 18 - 4x^2 - 12x - 9 = 0
   \]
   Simplify:
   \[
   -x^2 - 12x - 27 = 0
   \]
   Multiply the whole equation by \(-1\) to simplify:
   \[
   x^2 + 12x + 27 = 0
   \]

4. **Solve the quadratic equation:**

   Use the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   For \( a = 1 \), \( b = 12 \), and \( c = 27 \):
   \[
   x = \frac{-12 \pm \sqrt{12^2 - 4(1)(27)}}{2(1)}
   \]
   \[
   x = \frac{-12 \pm \sqrt{144 - 108}}{2}
   \]
   \[
   x = \frac{-12 \pm \sqrt{36}}{2}
   \]
   \[
   x = \frac{-12 \pm 6}{2}
   \]
   This gives two solutions:
   \[
   x = \frac{-12 + 6}{2} = \frac{-6}{2} = -3
   \]
   \[
   x = \frac{-12 - 6}{2} = \frac{-18}{2} = -9
   \]

5. **Check for restrictions:**

   The base of the logarithm, \( -2x - 3 \), must be positive. This gives the inequality:
   \[
   -2x - 3 > 0
   \]
   Solve for \( x \):
   \[
   -2x > 3
   \]
   \[
   x < -\frac{3}{2}
   \]
   So, \( x \) must be less than \( -\frac{3}{2} \). Checking our solutions:
   - \( x = -3 \) satisfies \( x < -\frac{3}{2} \),
   - \( x = -9 \) also satisfies \( x < -\frac{3}{2} \).

Thus, the solutions are \( x = -3 \) and \( x = -9 \).

Would you like further details or have any questions?

### Related Questions:
1. How is the exponential form derived from a logarithmic equation?
2. What is the quadratic formula and when is it used?
3. How can we verify solutions in logarithmic equations?
4. What is the domain of a logarithmic function with a negative base?
5. How do restrictions on the base of a logarithmic function affect the solution set?

### Tip:
Always check for domain restrictions when solving logarithmic equations to ensure all solutions are valid.

Solve for all values of x: log_{-2x-3}(3x^2 - 18) = 2

Learn how to solve the logarithmic equation log_{-2x-3}(3x^2 - 18) = 2 with a detailed step-by-step solution. This problem involves converting a logarithmic equation to exponential form and solving the resulting quadratic equation. Suitable for high school students in Grades 10-12.

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