Evaluate: ((3)/(x-x^2)+(1)/(1-x))/(9-x^2)/(1-x)

Expression: $\left( \frac{ 3 }{ x-{x}^{2} }+\frac{ 1 }{ 1-x } \right)\div\frac{ 9-{x}^{2} }{ 1-x }$

Factor out $x$ from the expression

$\left( \frac{ 3 }{ x \times \left( 1-x \right) }+\frac{ 1 }{ 1-x } \right)\div\frac{ 9-{x}^{2} }{ 1-x }$

To divide by a fraction, multiply by the reciprocal of that fraction

$\left( \frac{ 3 }{ x \times \left( 1-x \right) }+\frac{ 1 }{ 1-x } \right) \times \frac{ 1-x }{ 9-{x}^{2} }$

Write all numerators above the least common denominator $x \times \left( 1-x \right)$

$\frac{ 3+x }{ x \times \left( 1-x \right) } \times \frac{ 1-x }{ 9-{x}^{2} }$

Use ${a}^{2}-{b}^{2}=\left( a-b \right)\left( a+b \right)$ to factor the expression

$\frac{ 3+x }{ x \times \left( 1-x \right) } \times \frac{ 1-x }{ \left( 3-x \right) \times \left( 3+x \right) }$

Cancel out the common factor $3+x$

$\frac{ 1 }{ x \times \left( 1-x \right) } \times \frac{ 1-x }{ 3-x }$

Cancel out the common factor $1-x$

$\frac{ 1 }{ x } \times \frac{ 1 }{ 3-x }$

Multiply the fractions

$\frac{ 1 }{ x \times \left( 3-x \right) }$

Distribute $x$ through the parentheses

$\frac{ 1 }{ 3x-{x}^{2} }$