Solve x/x^2+2x-3+1/1-2x+x^2

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

Evaluate

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

Short Solution Steps

Factor x^{2}+2x-3. Factor 1-2x+x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+3\right) and \left(x-1\right)^{2} is \left(x+3\right)\left(x-1\right)^{2}. Multiply \frac{x}{\left(x-1\right)\left(x+3\right)} times \frac{x-1}{x-1}. Multiply \frac{1}{\left(x-1\right)^{2}} times \frac{x+3}{x+3}.

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

Since \frac{x\left(x-1\right)}{\left(x+3\right)\left(x-1\right)^{2}} and \frac{x+3}{\left(x+3\right)\left(x-1\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x\left(x-1\right)+x+3.

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

Combine like terms in x^{2}-x+x+3.

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

Expand \left(x+3\right)\left(x-1\right)^{2}.

$$\frac{x^{2}+3}{x^{3}+x^{2}-5x+3}$$

Differentiate w.r.t. x

$$-\frac{\left(x+1\right)\left(x^{2}+15\right)}{\left(x+3\right)^{2}\left(x-1\right)^{3}}$$