Find the value of each of the following if x+2 :  a) x+1/x-1 * x+3/1-x^2 / (x+3)^2/1-x   b) x^2+6x+5/x^2+7x+12 * x^2+2x-8/x^2-25 / x^2-x-2/x^2-2x-15

Let's solve the given expressions step by step, using the condition \(x = -2\).

### Part (a):
\[
\text{Simplify and find the value: } \frac{\frac{x+1}{x-1} \times \frac{x+3}{1-x^2}}{\frac{(x+3)^2}{1-x}}
\]

**Step 1:** Substitute \(x = -2\) into each part of the expression.

- For the first term \(\frac{x+1}{x-1}\):
  \[
  \frac{-2+1}{-2-1} = \frac{-1}{-3} = \frac{1}{3}
  \]

- For the second term \(\frac{x+3}{1-x^2}\):
  \[
  \frac{-2+3}{1-(-2)^2} = \frac{1}{1-4} = \frac{1}{-3} = -\frac{1}{3}
  \]

- For the third term \(\frac{(x+3)^2}{1-x}\):
  \[
  \frac{(-2+3)^2}{1-(-2)} = \frac{1^2}{1+2} = \frac{1}{3}
  \]

**Step 2:** Combine the values from each term:

\[
\frac{\frac{1}{3} \times -\frac{1}{3}}{\frac{1}{3}} = \frac{-\frac{1}{9}}{\frac{1}{3}} = -\frac{1}{9} \times \frac{3}{1} = -\frac{1}{3}
\]

So, the value of part (a) when \(x = -2\) is \(-\frac{1}{3}\).

### Part (b):
\[
\text{Simplify and find the value: } \frac{\frac{x^2+6x+5}{x^2+7x+12} \times \frac{x^2+2x-8}{x^2-25}}{\frac{x^2-x-2}{x^2-2x-15}}
\]

**Step 1:** Substitute \(x = -2\) into each part of the expression.

- For the first term \(\frac{x^2 + 6x + 5}{x^2 + 7x + 12}\):
  \[
  \frac{(-2)^2 + 6(-2) + 5}{(-2)^2 + 7(-2) + 12} = \frac{4 - 12 + 5}{4 - 14 + 12} = \frac{-3}{2}
  \]

- For the second term \(\frac{x^2 + 2x - 8}{x^2 - 25}\):
  \[
  \frac{(-2)^2 + 2(-2) - 8}{(-2)^2 - 25} = \frac{4 - 4 - 8}{4 - 25} = \frac{-8}{-21} = \frac{8}{21}
  \]

- For the third term \(\frac{x^2 - x - 2}{x^2 - 2x - 15}\):
  \[
  \frac{(-2)^2 - (-2) - 2}{(-2)^2 - 2(-2) - 15} = \frac{4 + 2 - 2}{4 + 4 - 15} = \frac{4}{-7} = -\frac{4}{7}
  \]

**Step 2:** Combine the values from each term:

\[
\frac{\frac{-3}{2} \times \frac{8}{21}}{-\frac{4}{7}} = \frac{-\frac{24}{42}}{-\frac{4}{7}} = \frac{-\frac{4}{7}}{-\frac{4}{7}} = 1
\]

So, the value of part (b) when \(x = -2\) is \(1\).

### Final Results:
- **(a)** When \(x = -2\), the value is \(-\frac{1}{3}\).
- **(b)** When \(x = -2\), the value is \(1\).

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Do you want further details or have any questions about these steps?

Here are 5 related questions:
1. How do you simplify expressions with multiple fractions?
2. What happens when you substitute a specific value into a complex algebraic expression?
3. How do you factor quadratic expressions?
4. What are the steps to simplify rational expressions?
5. How do you handle negative exponents and negative signs in algebraic fractions?

**Tip:** When substituting values into an expression, double-check your arithmetic, especially when dealing with negative numbers.

Learn how to solve complex rational expression equations by substituting values. This problem involves algebraic manipulation and substitution techniques. Suitable for high school students.

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