Classwork (Simplify $$ \frac{1}{x-4}+\frac{2}{x+3}-\frac{2}{x^{2}-x-12} $$ $$ 2 \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}+x-2} $$

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Calculate or simplify the expression $$ \frac{1}{x-4}+\frac{2}{x+3}-\frac{2}{x^{2}-x-12} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{1}{x-4}+\frac{2}{x+3}-\frac{2}{x^{2}-x-12}$$
- step1: Factor the expression:
  $$\frac{1}{x-4}+\frac{2}{x+3}-\frac{2}{\left(x-4ight)\left(x+3ight)}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{x+3}{\left(x-4ight)\left(x+3ight)}+\frac{2\left(x-4ight)}{\left(x+3ight)\left(x-4ight)}-\frac{2}{\left(x-4ight)\left(x+3ight)}$$
- step3: Rewrite the expression:
  $$\frac{x+3}{\left(x-4ight)\left(x+3ight)}+\frac{2\left(x-4ight)}{\left(x-4ight)\left(x+3ight)}-\frac{2}{\left(x-4ight)\left(x+3ight)}$$
- step4: Transform the expression:
  $$\frac{x+3+2\left(x-4ight)-2}{\left(x-4ight)\left(x+3ight)}$$
- step5: Multiply the terms:
  $$\frac{x+3+2x-8-2}{\left(x-4ight)\left(x+3ight)}$$
- step6: Calculate:
  $$\frac{3x-7}{\left(x-4ight)\left(x+3ight)}$$
- step7: Multiply the terms:
  $$\frac{3x-7}{x^{2}-x-12}$$
Calculate or simplify the expression $$ 2 \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}+x-2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(2	imes \frac{x}{x+2}ight)-\frac{2}{x-1}-\frac{6}{x^{2}+x-2}$$
- step1: Multiply the terms:
  $$\frac{2x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}+x-2}$$
- step2: Factor the expression:
  $$\frac{2x}{x+2}-\frac{2}{x-1}-\frac{6}{\left(x+2ight)\left(x-1ight)}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{2x\left(x-1ight)}{\left(x+2ight)\left(x-1ight)}-\frac{2\left(x+2ight)}{\left(x-1ight)\left(x+2ight)}-\frac{6}{\left(x+2ight)\left(x-1ight)}$$
- step4: Rewrite the expression:
  $$\frac{2x\left(x-1ight)}{\left(x+2ight)\left(x-1ight)}-\frac{2\left(x+2ight)}{\left(x+2ight)\left(x-1ight)}-\frac{6}{\left(x+2ight)\left(x-1ight)}$$
- step5: Transform the expression:
  $$\frac{2x\left(x-1ight)-2\left(x+2ight)-6}{\left(x+2ight)\left(x-1ight)}$$
- step6: Multiply the terms:
  $$\frac{2x^{2}-2x-2\left(x+2ight)-6}{\left(x+2ight)\left(x-1ight)}$$
- step7: Multiply the terms:
  $$\frac{2x^{2}-2x-\left(2x+4ight)-6}{\left(x+2ight)\left(x-1ight)}$$
- step8: Subtract the terms:
  $$\frac{2x^{2}-4x-10}{\left(x+2ight)\left(x-1ight)}$$
- step9: Multiply the terms:
  $$\frac{2x^{2}-4x-10}{x^{2}+x-2}$$
The simplified form of the first expression is $$\frac{3x-7}{x^{2}-x-12}$$ and the simplified form of the second expression is $$\frac{2x^{2}-4x-10}{x^{2}+x-2}$$.
Simplify the expressions: 1. $$ \frac{1}{x-4}+\frac{2}{x+3}-\frac{2}{x^{2}-x-12} $$ simplifies to $$ \frac{3x-7}{x^{2}-x-12} $$. 2. $$ 2 \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}+x-2} $$ simplifies to $$ \frac{2x^{2}-4x-10}{x^{2}+x-2} $$.

Classwork (Simplify \[ \frac{1}{x-4}+\frac{2}{x+3}-\frac{2}{x^{2}-x-12} \] \[ 2 \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}+x-2} \]

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