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The given equation is:

\[
\frac{7}{20}x + \frac{x - 2}{15} + \frac{1}{12}x^2 = \frac{1}{12}(x + 3)^2 - \frac{1}{20}(2x + 3)
\]

First, let's simplify and solve this equation step-by-step.

1. **Expand both sides of the equation:**

   \[
   \frac{1}{12}(x + 3)^2 = \frac{1}{12}(x^2 + 6x + 9)
   \]
   \[
   \frac{1}{12}(x^2 + 6x + 9) = \frac{1}{12}x^2 + \frac{6}{12}x + \frac{9}{12}
   \]
   \[
   = \frac{1}{12}x^2 + \frac{1}{2}x + \frac{3}{4}
   \]

   \[
   \frac{1}{20}(2x + 3) = \frac{2}{20}x + \frac{3}{20} = \frac{1}{10}x + \frac{3}{20}
   \]

   So, the equation becomes:

   \[
   \frac{7}{20}x + \frac{x - 2}{15} + \frac{1}{12}x^2 = \frac{1}{12}x^2 + \frac{1}{2}x + \frac{3}{4} - \left(\frac{1}{10}x + \frac{3}{20}\right)
   \]

2. **Combine like terms on the right-hand side:**

   \[
   \frac{1}{2}x - \frac{1}{10}x = \frac{5}{10}x - \frac{1}{10}x = \frac{4}{10}x = \frac{2}{5}x
   \]
   \[
   \frac{3}{4} - \frac{3}{20} = \frac{15}{20} - \frac{3}{20} = \frac{12}{20} = \frac{3}{5}
   \]

   Now the equation is:

   \[
   \frac{7}{20}x + \frac{x - 2}{15} + \frac{1}{12}x^2 = \frac{1}{12}x^2 + \frac{2}{5}x + \frac{3}{5}
   \]

3. **Subtract \(\frac{1}{12}x^2\) from both sides:**

   \[
   \frac{7}{20}x + \frac{x - 2}{15} = \frac{2}{5}x + \frac{3}{5}
   \]

4. **Combine the x terms on the left-hand side:**

   Convert all terms to a common denominator. The common denominator of 20, 15, and 5 is 60.

   \[
   \frac{7}{20}x = \frac{21}{60}x
   \]
   \[
   \frac{x - 2}{15} = \frac{4(x - 2)}{60} = \frac{4x - 8}{60}
   \]
   \[
   \frac{2}{5}x = \frac{24}{60}x
   \]
   \[
   \frac{3}{5} = \frac{36}{60}
   \]

   So the equation becomes:

   \[
   \frac{21}{60}x + \frac{4x - 8}{60} = \frac{24}{60}x + \frac{36}{60}
   \]

   Combine like terms:

   \[
   \frac{21x + 4x - 8}{60} = \frac{24x + 36}{60}
   \]

   Simplify:

   \[
   \frac{25x - 8}{60} = \frac{24x + 36}{60}
   \]

   Multiply both sides by 60 to clear the denominator:

   \[
   25x - 8 = 24x + 36
   \]

5. **Solve for x:**

   \[
   25x - 24x = 36 + 8
   \]
   \[
   x = 44
   \]

Therefore, the solution to the equation is \( x = 44 \).

Would you like any further details or have any other questions?

Here are 5 more questions you might ask:
1. Can you solve another algebraic equation for me?
2. How do you solve quadratic equations?
3. Can you help me with systems of equations?
4. What is the process for solving inequalities?
5. Can you explain how to factor polynomials?

**Tip:** Always check your solution by substituting it back into the original equation to verify that it satisfies the equation.

Learn how to resolve this complex fractional equation step-by-step. Includes detailed simplification and solution process. Suitable for high school students and those preparing for advanced algebraic challenges.

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