(5)/(2x^(3))-(3x-9)/(x^(2)-6x+9)+(12 x)/(4x^(2)-12 x)

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Factor Denominators: Factor the denominators where possible. x^2 - 6x + 9 = (x - 3)^2 4x^2 - 12x = 4x(x - 3)Rewrite Expression: Rewrite the expression with factored denominators. (5)/(2x^3) - (3x - 9)/((x - 3)^2) + (12x)/(4x(x - 3))Simplify Third Term: Simplify the third term by canceling out common factors. (12x)/(4x(x - 3)) = 3/(x - 3)Find Common Denominator: Find a common denominator for all terms, which is 2x^3(x - 3)^2.Rewrite Over Common Denominator: Rewrite each term over the common denominator. (5)/(2x^3) becomes (5(x - 3)^2)/(2x^3(x - 3)^2) (3x - 9)/((x - 3)^2) becomes (3x - 9)(2x^3)/(2x^3(x - 3)^2) (3)/(x - 3) becomes (3(2x^3))/(2x^3(x - 3)^2)Combine Over Common Denominator: Combine the terms over the common denominator. (5(x - 3)^2 - (3x - 9)(2x^3) + 3(2x^3))/(2x^3(x - 3)^2)Expand and Simplify Numerator: Expand and simplify the numerator. 5(x^2 - 6x + 9) - 6x^4 + 18x^3 + 6x^3Combine Like Terms in Numerator: Combine like terms in the numerator. 5x^2 - 30x + 45 - 6x^4 + 18x^3 + 6x^3Continue Combining Like Terms: Continue combining like terms. -6x^4 + (18x^3 + 6x^3) + (5x^2 - 30x) + 45Finish Combining Like Terms: Finish combining like terms. -6x^4 + 24x^3 + 5x^2 - 30x + 45Write Simplified Expression: Write the simplified expression. (-6x^4 + 24x^3 + 5x^2 - 30x + 45)/(2x^3(x - 3)^2)

$\frac{5}{2 x^{3}}-\frac{3 x-9}{x^{2}-6 x+9}+\frac{12 x}{4 x^{2}-12 x}$

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$\frac{5}{2 x^{3}}-\frac{3 x-9}{x^{2}-6 x+9}+\frac{12 x}{4 x^{2}-12 x}$