Find the complete set of real values of a for which both roots of the quadratic equation `(a^(2)-6a+5)x^(2)-sqrt(a^(2)+2a)x+(6a-a^(2)-8)=0` lie on either side of the origin.

Answer

To solve the quadratic equation \( (a^2 - 6a + 5)x^2 - \sqrt{a^2 + 2a}x + (6a - a^2 - 8) = 0 \) and find the complete set of real values of \( a \) for which both roots lie on either side of the origin, we follow these steps:

### Step 1: Identify the coefficients
The quadratic equation is in the form \( Ax^2 + Bx + C = 0 \) where:
- \( A = a^2 - 6a + 5 \)
- \( B = -\sqrt{a^2 + 2a} \)
- \( C = 6a - a^2 - 8 \)

### Step 2: Condition for real roots
For the roots to be real, the discriminant \( D \) must be non-negative:
\[ D = B^2 - 4AC \geq 0 \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = (-\sqrt{a^2 + 2a})^2 - 4(a^2 - 6a + 5)(6a - a^2 - 8) \geq 0 \]
This simplifies to:
\[ a^2 + 2a - 4(a^2 - 6a + 5)(6a - a^2 - 8) \geq 0 \]

### Step 3: Condition for roots on either side of the origin
For the roots to lie on either side of the origin, the product of the roots \( \frac{C}{A} < 0 \):
\[ \frac{6a - a^2 - 8}{a^2 - 6a + 5} < 0 \]

### Step 4: Analyze the signs of \( A \) and \( C \)
1. **Find the roots of \( A \)**:
\[ a^2 - 6a + 5 = 0 \implies (a - 1)(a - 5) = 0 \implies a = 1, 5 \]
The sign of \( A \) changes at \( a = 1 \) and \( a = 5 \).

2. **Find the roots of \( C \)**:
\[ 6a - a^2 - 8 = 0 \implies -a^2 + 6a - 8 = 0 \implies (a - 2)(a - 4) = 0 \implies a = 2, 4 \]
The sign of \( C \) changes at \( a = 2 \) and \( a = 4 \).

### Step 5: Determine intervals
We need to analyze the intervals defined by the roots \( 1, 2, 4, 5 \):
- Test intervals: \( (-\infty, 1) \), \( (1, 2) \), \( (2, 4) \), \( (4, 5) \), \( (5, \infty) \).

### Step 6: Sign analysis
1. **Interval \( (-\infty, 1) \)**: \( A > 0 \), \( C < 0 \) → \( \frac{C}{A} < 0 \) (valid)
2. **Interval \( (1, 2) \)**: \( A < 0 \), \( C < 0 \) → \( \frac{C}{A} > 0 \) (invalid)
3. **Interval \( (2, 4) \)**: \( A < 0 \), \( C > 0 \) → \( \frac{C}{A} < 0 \) (valid)
4. **Interval \( (4, 5) \)**: \( A < 0 \), \( C < 0 \) → \( \frac{C}{A} > 0 \) (invalid)
5. **Interval \( (5, \infty) \)**: \( A > 0 \), \( C > 0 \) → \( \frac{C}{A} > 0 \) (invalid)

### Step 7: Combine valid intervals
The valid intervals where both roots lie on either side of the origin are:
\[ (-\infty, 1) \cup (2, 4) \]

### Final Answer
The complete set of real values of \( a \) for which both roots lie on either side of the origin is:
\[ a \in (-\infty, 1) \cup (2, 4) \]