a) (1-x^2)/(x^2)+2x:(2-2x)/(3x)

To solve the expression \(\frac-x^{2}}{x^{2}+2x}:\frac {2-2x}{3x}\), we need to follow these steps:<br /><br />1. **Rewrite the division as multiplication by the reciprocal:**<br /> \[<br /> \frac {1-x^{2}}{x^{2}+2x} \div \frac {2-2x}{3x} = \frac {1-x^{2}}{x^{2}+2x} \times \frac {3x}{2-2x}<br /> \]<br /><br />2. **Factorize the numerators and denominators where possible:**<br /> \[<br /> 1 - x^2 = (1 - x)(1 + x)<br /> \]<br /> \[<br /> x^2 + 2x = x(x + 2)<br /> \]<br /> \[<br /> 2 - 2x = 2(1 - x)<br /> \]<br /><br />3. **Substitute the factored forms into the expression:**<br /> \[<br /> \frac{(1 - x)(1 + x)}{x(x + 2)} \times \frac{3x}{2(1 - x)}<br /> \]<br /><br />4. **Cancel out common factors in the numerator and denominator:**<br /> \[<br /> \frac{(1 - x)(1 + x)}{x(x + 2)} \times \frac{3x}{2(1 - x)} = \frac{(1 + x)}{x(x + 2)} \times \frac{3x}{2}<br /> \]<br /> Here, \((1 - x)\) cancels out.<br /><br />5. **Simplify the remaining expression:**<br /> \frac{1 + x}{x(x + 2)} \times \frac{3x}{2} = \frac{(1 + x) \cdot 3x}{x(x + 2) \cdot 2}<br /> \]<br /> \[<br /> = \frac{3x(1 + x)}{2x(x + 2)}<br /> \]<br /> \[<br /> = \frac{3(1 + x)}{2(x + 2)}<br /> \]<br /><br />6. **Further simplify if possible:**<br /> \[<br /> = \frac{3(1 + x)}{2(x + 2)}<br /> \]<br /><br />Thus, the simplified form of the given expression is:<br />\[<br />\boxed{\frac{3(1 + x)}{2(x + 2)}}<br />\