Solve x^2-5x/x-1-x^2-25/x^2+x+5

$$\frac { x ^ { 2 } - 5 x } { x - 1 } - \frac { x ^ { 2 } - 25 } { x ^ { 2 } + x + 5 }$$

Evaluate

$$\frac{\left(x-5\right)\left(x^{3}+x+5\right)}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x^{2}+x+5 is \left(x-1\right)\left(x^{2}+x+5\right). Multiply \frac{x^{2}-5x}{x-1} times \frac{x^{2}+x+5}{x^{2}+x+5}. Multiply \frac{x^{2}-25}{x^{2}+x+5} times \frac{x-1}{x-1}.

$$\frac{\left(x^{2}-5x\right)\left(x^{2}+x+5\right)}{\left(x-1\right)\left(x^{2}+x+5\right)}-\frac{\left(x^{2}-25\right)\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Since \frac{\left(x^{2}-5x\right)\left(x^{2}+x+5\right)}{\left(x-1\right)\left(x^{2}+x+5\right)} and \frac{\left(x^{2}-25\right)\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}-5x\right)\left(x^{2}+x+5\right)-\left(x^{2}-25\right)\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Do the multiplications in \left(x^{2}-5x\right)\left(x^{2}+x+5\right)-\left(x^{2}-25\right)\left(x-1\right).

$$\frac{x^{4}+x^{3}+5x^{2}-5x^{3}-5x^{2}-25x-x^{3}+x^{2}+25x-25}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Combine like terms in x^{4}+x^{3}+5x^{2}-5x^{3}-5x^{2}-25x-x^{3}+x^{2}+25x-25.

$$\frac{x^{4}-5x^{3}+x^{2}-25}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Expand \left(x-1\right)\left(x^{2}+x+5\right).

$$\frac{x^{4}-5x^{3}+x^{2}-25}{x^{3}+4x-5}$$

Expand

$$\frac{x^{4}-5x^{3}+x^{2}-25}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x^{2}+x+5 is \left(x-1\right)\left(x^{2}+x+5\right). Multiply \frac{x^{2}-5x}{x-1} times \frac{x^{2}+x+5}{x^{2}+x+5}. Multiply \frac{x^{2}-25}{x^{2}+x+5} times \frac{x-1}{x-1}.

$$\frac{\left(x^{2}-5x\right)\left(x^{2}+x+5\right)}{\left(x-1\right)\left(x^{2}+x+5\right)}-\frac{\left(x^{2}-25\right)\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Since \frac{\left(x^{2}-5x\right)\left(x^{2}+x+5\right)}{\left(x-1\right)\left(x^{2}+x+5\right)} and \frac{\left(x^{2}-25\right)\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}-5x\right)\left(x^{2}+x+5\right)-\left(x^{2}-25\right)\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Do the multiplications in \left(x^{2}-5x\right)\left(x^{2}+x+5\right)-\left(x^{2}-25\right)\left(x-1\right).

$$\frac{x^{4}+x^{3}+5x^{2}-5x^{3}-5x^{2}-25x-x^{3}+x^{2}+25x-25}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Combine like terms in x^{4}+x^{3}+5x^{2}-5x^{3}-5x^{2}-25x-x^{3}+x^{2}+25x-25.

$$\frac{x^{4}-5x^{3}+x^{2}-25}{\left(x-1\right)\left(x^{2}+x+5\right)}$$

Expand \left(x-1\right)\left(x^{2}+x+5\right).

$$\frac{x^{4}-5x^{3}+x^{2}-25}{x^{3}+4x-5}$$