3/(2-x)^2-5/(x+2)^2=14/x^2-4

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To solve this equation, we first need to find a common denominator for all the fractions. The common denominator is the product of the denominators of each fraction: (2-x)^2 (x+2)^2 x^2.

We then multiply each term by the necessary factors to clear the denominators:

(3 (x+2)^2 x^2) - (5 (2-x)^2 x^2) = 14(2-x)^2(x+2)^2

Expanding the expressions:

3(x^2 + 4x + 4)(x^2) - 5(4 - 4x + x^2)(x^2) = 14(4 - 4x + x^2)(x + 2)^2

3(x^4 + 4x^3 + 4x^2) - 5(4x^2 - 4x^3 + x^4) = 14(4 - 4x + x^2)(x^2 + 4x + 4)

Now we simplify the equation:

3x^4 + 12x^3 + 12x^2 - 20x^2 + 20x^3 - 5x^4 = 14(x^2 - 4x + 4)(x^2 + 4x + 4)

-2x^4 + 32x^3 - 8x^2 = 14(x^4 + 4x^2 + 16x^2 - 16x + 4x - 16)

Simplify further:

-2x^4 + 32x^3 - 8x^2 = 14(x^4 + 20x^2 - 16x - 16)

Multiplying out:

-2x^4 + 32x^3 - 8x^2 = 14x^4 + 280x^2 - 224x - 224

Rearranging terms:

0 = 16x^4 - 32x^3 + 288x^2 - 224x - 224

This is a quartic equation that can be solved using various methods or software to find the possible values of x.