(b) Simplify : $$ \frac{x+1}{x^{2}+5 x+6}-\frac{x}{x+3} $$

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Factor the expression by following steps:
- step0: Factor:
  $$x^{2}+5x+6$$
- step1: Rewrite the expression:
  $$x^{2}+\left(3+2\right)x+6$$
- step2: Calculate:
  $$x^{2}+3x+2x+6$$
- step3: Rewrite the expression:
  $$x\times x+x\times 3+2x+2\times 3$$
- step4: Factor the expression:
  $$x\left(x+3\right)+2\left(x+3\right)$$
- step5: Factor the expression:
  $$\left(x+2\right)\left(x+3\right)$$
Calculate or simplify the expression $$ \frac{x+1}{x^{2}+5 x+6}-\frac{x}{x+3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{x+1}{x^{2}+5x+6}-\frac{x}{x+3}$$
- step1: Factor the expression:
  $$\frac{x+1}{\left(x+2\right)\left(x+3\right)}-\frac{x}{x+3}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{x+1}{\left(x+2\right)\left(x+3\right)}-\frac{x\left(x+2\right)}{\left(x+3\right)\left(x+2\right)}$$
- step3: Rewrite the expression:
  $$\frac{x+1}{\left(x+2\right)\left(x+3\right)}-\frac{x\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}$$
- step4: Transform the expression:
  $$\frac{x+1-x\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}$$
- step5: Multiply the terms:
  $$\frac{x+1-\left(x^{2}+2x\right)}{\left(x+2\right)\left(x+3\right)}$$
- step6: Calculate:
  $$\frac{-x+1-x^{2}}{\left(x+2\right)\left(x+3\right)}$$
- step7: Multiply the terms:
  $$\frac{-x+1-x^{2}}{x^{2}+5x+6}$$
To simplify the expression $$ \frac{x+1}{x^{2}+5 x+6}-\frac{x}{x+3} $$, we can follow these steps:

1. **Factor the denominator of the first fraction**:
   The expression $$ x^{2}+5x+6 $$ can be factored as:
   $$
   (x+2)(x+3)
   $$

2. **Rewrite the expression**:
   Now we can rewrite the original expression using the factored form:
   $$
   \frac{x+1}{(x+2)(x+3)} - \frac{x}{x+3}
   $$

3. **Find a common denominator**:
   The common denominator for the two fractions is $$ (x+2)(x+3) $$. We can rewrite the second fraction:
   $$
   \frac{x}{x+3} = \frac{x(x+2)}{(x+2)(x+3)}
   $$

4. **Combine the fractions**:
   Now we can combine the two fractions:
   $$
   \frac{x+1 - x(x+2)}{(x+2)(x+3)}
   $$

5. **Simplify the numerator**:
   The numerator simplifies as follows:
   $$
   x + 1 - (x^2 + 2x) = -x^2 - x + 1
   $$

6. **Final expression**:
   Thus, the simplified expression is:
   $$
   \frac{-x^2 - x + 1}{(x+2)(x+3)}
   $$

So, the final simplified form of the expression is:
$$
\frac{-x^2 - x + 1}{(x+2)(x+3)}
$$
$$ \frac{-x^2 - x + 1}{(x+2)(x+3)} $$

(b) Simplify :

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