(16 / x^2-16) + (x / x+4) = 2 / x-4

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To solve this equation, we first need to find a common denominator for all the fractions. In this case, the common denominator is (x^2 - 16)(x + 4)(x - 4).

Rewriting each fraction with the common denominator:

(16 / x^2 - 16) = 16(x + 4)(x - 4) / (x^2 - 16)(x + 4)(x - 4
(x / x + 4) = x(x^2 - 16) / (x^2 - 16)(x + 4)(x - 4
(2 / x - 4) = 2(x^2 - 16) / (x^2 - 16)(x + 4)(x - 4)

Now, we can rewrite the equation with the common denominator:

16(x + 4)(x - 4) / (x^2 - 16)(x + 4)(x - 4) + x(x^2 - 16) / (x^2 - 16)(x + 4)(x - 4) = 2(x^2 - 16) / (x^2 - 16)(x + 4)(x - 4)

Combining the fractions on the left side:

[16(x + 4)(x - 4) + x(x^2 - 16)] / (x^2 - 16)(x + 4)(x - 4) = 2(x^2 - 16) / (x^2 - 16)(x + 4)(x - 4)

Expanding and simplifying the numerators:

[16(x^2 - 16) + x(x^2 - 16)] / (x^2 - 16)(x + 4)(x - 4) = 2(x^2 - 16) / (x^2 - 16)(x + 4)(x - 4)

[16x^2 - 256 + x^3 - 16x] / (x^2 - 16)(x + 4)(x - 4) = 2(x^2 - 16) / (x^2 - 16)(x + 4)(x - 4)

Combining like terms:

(x^3 + 16x^2 - 16x - 256) / (x^2 - 16)(x + 4)(x - 4) = 2(x^2 - 16) / (x^2 - 16)(x + 4)(x - 4)

Now we have a single fraction on both sides of the equation. To solve it, we can cancel out (x^2 - 16)(x + 4)(x - 4) from both sides:

x^3 + 16x^2 - 16x - 256 = 2(x^2 - 16)

Simplify the right side:

x^3 + 16x^2 - 16x - 256 = 2x^2 - 32

Combine like terms:

x^3 + 14x^2 - 16x - 224 = 0

This is the simplified form of the equation. It may not have a simple algebraic solution, so the final answer is x^3 + 14x^2 - 16x - 224 = 0.