Find the compounded ratio of:
`(x^(2)-25): (x^(2) + 3x-10), (x^(2)-4): (x^(2) + 3x + 2) and (x+1): (x^(2) + 2x)`

Answer

To find the compounded ratio of the given expressions, we will follow these steps:

### Step 1: Identify the ratios
The ratios given are:
1. \( (x^2 - 25) : (x^2 + 3x - 10) \)
2. \( (x^2 - 4) : (x^2 + 3x + 2) \)
3. \( (x + 1) : (x^2 + 2x) \)

### Step 2: Write the compounded ratio formula
The compounded ratio of three ratios \( a : b, c : d, e : f \) is given by:
\[ \text{Compounded Ratio} = \frac{a \cdot c \cdot e}{b \cdot d \cdot f} \]

### Step 3: Substitute the expressions into the formula
Using the identified ratios:
\[ \text{Compounded Ratio} = \frac{(x^2 - 25)(x^2 - 4)(x + 1)}{(x^2 + 3x - 10)(x^2 + 3x + 2)(x^2 + 2x)} \]

### Step 4: Factor the expressions
1. **Factor \( x^2 - 25 \)**:
\[ x^2 - 25 = (x - 5)(x + 5) \quad \text{(Difference of squares)} \]

2. **Factor \( x^2 - 4 \)**:
\[ x^2 - 4 = (x - 2)(x + 2) \quad \text{(Difference of squares)} \]

3. **Factor \( x^2 + 3x - 10 \)**:
- We need two numbers that multiply to \(-10\) and add to \(3\). The numbers are \(5\) and \(-2\).
\[ x^2 + 3x - 10 = (x + 5)(x - 2) \]

4. **Factor \( x^2 + 3x + 2 \)**:
- We need two numbers that multiply to \(2\) and add to \(3\). The numbers are \(2\) and \(1\).
\[ x^2 + 3x + 2 = (x + 2)(x + 1) \]

5. **Factor \( x^2 + 2x \)**:
\[ x^2 + 2x = x(x + 2) \]

### Step 5: Substitute the factored forms back into the compounded ratio
Now substituting the factored forms:
\[ \text{Compounded Ratio} = \frac{(x - 5)(x + 5)(x - 2)(x + 2)(x + 1)}{(x + 5)(x - 2)(x + 2)(x + 1)(x)(x + 2)} \]

### Step 6: Cancel out common factors
We can cancel out the common factors:
- \( (x + 5) \)
- \( (x - 2) \)
- \( (x + 2) \)
- \( (x + 1) \)

After canceling, we have:
\[ \text{Compounded Ratio} = \frac{(x - 5)}{x} \]

### Step 7: Final expression
Thus, the compounded ratio is:
\[ \text{Compounded Ratio} = \frac{x - 5}{x} \]