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f)
(7x^(2)-13 x)/(25x^(2)-9):((2)/(5x^(2)-3x)-(3)/(2x)-(4)/(3-5x))

f) $\frac{7 x^{2}-13 x}{25 x^{2}-9}:\left(\frac{2}{5 x^{2}-3 x}-\frac{3}{2 x}-\frac{4}{3-5 x}\right)$

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Full solution

Q. f)

\frac{7 x^{2}-13 x}{25 x^{2}-9}:\left(\frac{2}{5 x^{2}-3 x}-\frac{3}{2 x}-\frac{4}{3-5 x}\right)

Find Common Denominator: Step $1$ : Simplify the complex fraction by finding a common denominator for the smaller fractions in the denominator of the main fraction. $\newline$ $\frac{2}{5x^2-3x} - \frac{3}{2x} - \frac{4}{3-5x}$ $\newline$ Common denominator = $2x(5x^2-3x)(3-5x)$
Rewrite Fractions: Step $2$ : Rewrite each fraction over the common denominator. $\newline$ $(2)(2x)(3-5x) / [2x(5x^2-3x)(3-5x)] - (3)(5x^2-3x)(3-5x) / [2x(5x^2-3x)(3-5x)] - (4)(2x)(5x^2-3x) / [2x(5x^2-3x)(3-5x)]$
Combine Numerators: Step $3$ : Combine the numerators over the common denominator. $\frac{4x(3-5x) - 3(5x^2-3x)(3-5x) - 8x(5x^2-3x)}{2x(5x^2-3x)(3-5x)}$
Simplify Numerator: Step $4$ : Simplify the numerator. $\newline$ $4x(3-5x) - 3(5x^2-3x)(3-5x) - 8x(5x^2-3x)$ $\newline$ = $12x - 20x^2 - 45x^3 + 27x^2 + 40x^3 - 24x^2$ $\newline$ = $-5x^3 + 3x^2 + 12x$
Place Over Common Denominator: Step $5$ : Place the simplified numerator over the common denominator. $\newline$ $(-5x^3 + 3x^2 + 12x) / [2x(5x^2-3x)(3-5x)]$
Simplify Main Fraction: Step $6$ : Simplify the main fraction by dividing the numerator of the main fraction by the denominator expression we just simplified. $\newline$ (\(7x^ $2$ - $13$ x) / [( $-5$ x^ $3$ + $3$ x^ $2$ + $12$ x) / ( $2$ x( $5$ x^ $2$ $-3$ x)( $3$ $-5$ x))] = ( $7$ x^ $2$ - $13$ x) * [ $2$ x( $5$ x^ $2$ $-3$ x)( $3$ $-5$ x) / ( $-5$ x^ $3$ + $3$ x^ $2$ + $12$ x)]
Cancel Common Factors: Step $7$ : Cancel out common factors and simplify. $\newline$ $= (7x^2 - 13x) \cdot \left[\frac{2x(5x^2-3x)(3-5x)}{-5x^3 + 3x^2 + 12x}\right]$ $\newline$ $= (7x - 13) \cdot \left[\frac{2(5x^2-3x)(3-5x)}{-5x + 3 + 12}\right]$ $\newline$ $= (7x - 13) \cdot \left[\frac{2(5x^2-3x)(3-5x)}{-5x + 15}\right]$

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