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The equation is:

\[
\frac{12}{x^2 + 8x + 16} - \frac{1}{x^2 - 8x + 16} + \frac{1}{x^2 - 16} = 0
\]

### Solution:

1. **Factor the Denominators**:
   - \( x^2 + 8x + 16 = (x + 4)^2 \)
   - \( x^2 - 8x + 16 = (x - 4)^2 \)
   - \( x^2 - 16 = (x - 4)(x + 4) \)

   So the equation becomes:
   
   \[
   \frac{12}{(x+4)^2} - \frac{1}{(x-4)^2} + \frac{1}{(x-4)(x+4)} = 0
   \]

2. **Common Denominator**:
   The common denominator is \( (x+4)^2(x-4)^2 \).

3. **Rewrite Each Term with the Common Denominator**:
   - \( \frac{12}{(x+4)^2} = \frac{12(x-4)^2}{(x+4)^2(x-4)^2} \)
   - \( \frac{1}{(x-4)^2} = \frac{(x+4)^2}{(x+4)^2(x-4)^2} \)
   - \( \frac{1}{(x-4)(x+4)} = \frac{(x-4)(x+4)}{(x+4)^2(x-4)^2} = \frac{x^2 - 16}{(x+4)^2(x-4)^2} \)

   So the equation is:

   \[
   \frac{12(x-4)^2 - (x+4)^2 + (x^2 - 16)}{(x+4)^2(x-4)^2} = 0
   \]

4. **Simplify the Numerator**:
   Now we simplify the numerator:

   - Expand \( 12(x-4)^2 = 12(x^2 - 8x + 16) = 12x^2 - 96x + 192 \)
   - Expand \( (x+4)^2 = x^2 + 8x + 16 \)

   The numerator becomes:

   \[
   12x^2 - 96x + 192 - (x^2 + 8x + 16) + (x^2 - 16)
   \]

   Simplify further:

   \[
   12x^2 - 96x + 192 - x^2 - 8x - 16 + x^2 - 16 = 12x^2 - x^2 + x^2 - 96x - 8x + 192 - 16 - 16
   \]
   
   \[
   = 12x^2 - 96x + 192 - x^2 - 8x - 16 + x^2 - 16 = 12x^2 - 96x + 
   \[
   -9x

12 / (x^2 + 8x + 16) - 1 / (x^2 - 8x + 16) + 1 / (x^2 - 16) = 0

This problem involves solving a rational equation by factoring quadratic expressions and finding a common denominator. Explore key algebraic concepts like factoring and simplifying rational expressions. Suitable for students in Grades 10-12.

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