Express each of the following as a rational expression.
`(x^(2)-5x+6)/(x^(2)-9x+20)+(x^(2)-3x+2)/(x^(2)-5x+4):`

Options

A. `((x-4))/((x-3))`
B. `((x-3))/((x-5))`
C. `((x-2))/((x-5))`
D. `((x-5))/((x-3))`

Answer

To express the given expression as a rational expression, we will follow these steps:

### Step 1: Factor the Quadratic Expressions
We need to factor each of the quadratic expressions in the given rational expression.

1. **Factor \(x^2 - 5x + 6\)**:
- We look for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)).
- The numbers \(-2\) and \(-3\) work since \(-2 \times -3 = 6\) and \(-2 + -3 = -5\).
- Thus, \(x^2 - 5x + 6 = (x - 2)(x - 3)\).

2. **Factor \(x^2 - 9x + 20\)**:
- We look for two numbers that multiply to \(20\) and add to \(-9\).
- The numbers \(-4\) and \(-5\) work since \(-4 \times -5 = 20\) and \(-4 + -5 = -9\).
- Thus, \(x^2 - 9x + 20 = (x - 4)(x - 5)\).

3. **Factor \(x^2 - 3x + 2\)**:
- We look for two numbers that multiply to \(2\) and add to \(-3\).
- The numbers \(-1\) and \(-2\) work since \(-1 \times -2 = 2\) and \(-1 + -2 = -3\).
- Thus, \(x^2 - 3x + 2 = (x - 1)(x - 2)\).

4. **Factor \(x^2 - 5x + 4\)**:
- We look for two numbers that multiply to \(4\) and add to \(-5\).
- The numbers \(-1\) and \(-4\) work since \(-1 \times -4 = 4\) and \(-1 + -4 = -5\).
- Thus, \(x^2 - 5x + 4 = (x - 1)(x - 4)\).

### Step 2: Rewrite the Original Expression
Now we can rewrite the original expression with the factored forms:

\[ \frac{(x - 2)(x - 3)}{(x - 4)(x - 5)} + \frac{(x - 1)(x - 2)}{(x - 1)(x - 4)} \]

### Step 3: Simplify the Expression
Next, we simplify the expression:

1. **Cancel Common Factors**:
- In the second term, \((x - 1)\) cancels out:
\[ \frac{(x - 2)(x - 3)}{(x - 4)(x - 5)} + \frac{(x - 2)}{(x - 4)} \]

2. **Find a Common Denominator**:
- The common denominator for both fractions is \((x - 4)(x - 5)\).
- Rewrite the second fraction with the common denominator:
\[ \frac{(x - 2)(x - 3)}{(x - 4)(x - 5)} + \frac{(x - 2)(x - 5)}{(x - 4)(x - 5)} \]

3. **Combine the Fractions**:
- Combine the numerators:
\[ \frac{(x - 2)(x - 3) + (x - 2)(x - 5)}{(x - 4)(x - 5)} \]

4. **Factor Out Common Terms**:
- Factor out \((x - 2)\) from the numerator:
\[ \frac{(x - 2)((x - 3) + (x - 5))}{(x - 4)(x - 5)} \]

5. **Simplify the Numerator**:
- Combine the terms in the parentheses:
\[ (x - 3) + (x - 5) = 2x - 8 \]
- Thus, we have:
\[ \frac{(x - 2)(2x - 8)}{(x - 4)(x - 5)} \]

### Step 4: Final Simplification
1. **Factor the Numerator**:
- Factor out \(2\) from the numerator:
\[ \frac{2(x - 2)(x - 4)}{(x - 4)(x - 5)} \]

2. **Cancel Common Factors**:
- Cancel \((x - 4)\):
\[ \frac{2(x - 2)}{(x - 5)} \]

### Final Answer
The final expression as a rational expression is:
\[ \frac{2(x - 2)}{(x - 5)} \]