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Express rational functions as a sum of fractions with simpler denominators. You can apply this to telescoping series to mass cancel terms in a seemingly complicated sum.

Convert \[ \frac{ 4x^4 + 12 x^2 + 8 x + 3 }{ x^3 + x } \] into partial fractions.

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Convert \[ \frac{ 2x^4 +2 x^3 +2 x^2 +7 x -5 } { x^2-x } \] into partial fractions.

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Convert \[ \frac{ 2 x^3 - 12 x^2 + 20x -3 } { x^2 - 6x + 9 } \] into partial fractions.

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Convert \[ \frac{ 3 x^3 -17 x^2 +32x -28 } { x^3 - 8x^2 + 20 x - 16 } \] into partial fractions.

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