[tex]\frac{1}{x^{2} }+\frac{1}{(2+x)^{2} }=10/9[/tex]

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To solve this equation, we need to first find a common denominator for the two fractions on the left side of the equation. The common denominator is [tex]9x^{2}(2+x)^{2}[/tex].

So the equation becomes:

[tex]\frac{9(2+x)^{2} + 9x^{2}}{9x^{2}(2+x)^{2}} = \frac{10}{9}[/tex]

Simplifying the numerator:

[tex]9(4 + 4x + x^{2} + x^{2}) = 10 \cdot 9x^{2}(2+x)^{2}[/tex]

[tex]9(2x^{2} + 4x + 4) = 10 \cdot 9x^{2}(2+x)^{2}[/tex]

[tex]18x^{2} + 36x + 36 = 10 \cdot 9x^{2}(2 + x)^{2}[/tex]

[tex]18x^{2} + 36x + 36 = 90x^{2}(4 + 4x + x^{2})[/tex]

[tex]18x^{2} + 36x + 36 = 360x^{2} + 360x^{3} + 90x^{2}[/tex]

[tex]360x^{3} + 108x^{2} + 324x - 36 = 0[/tex]

Dividing by 36:

[tex]10x^{3} + 3x^{2} + 9x - 1 = 0[/tex]

Unfortunately, this equation is not easily factorable or solvable by simple algebraic manipulations, so it might need to be solved using numerical methods or a computer algebra system.