Solve each of the following quadratic equations:
`x^(2)+5x-(a^(2)+a-6)=0`

Explanation

To solve the quadratic equation \( x^2 + 5x - (a^2 + a - 6) = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
The equation is already in the standard form of a quadratic equation, which is \( Ax^2 + Bx + C = 0 \). Here, we can identify:
- \( A = 1 \)
- \( B = 5 \)
- \( C = -(a^2 + a - 6) \)

### Step 2: Simplify the constant term
We can simplify \( C \):
\[ C = -a^2 - a + 6 \]
So the equation becomes:
\[ x^2 + 5x + (-a^2 - a + 6) = 0 \]

### Step 3: Factor the quadratic equation
To factor this quadratic equation, we need to find two numbers that multiply to \( C \) (which is \(-a^2 - a + 6\)) and add up to \( B \) (which is \(5\)).

We can rewrite the equation as:
\[ x^2 + 5x + (-a^2 - a + 6) = 0 \]

### Step 4: Use the quadratic formula
If factoring is not straightforward, we can use the quadratic formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-a^2 - a + 6)}}{2 \cdot 1} \]

### Step 5: Calculate the discriminant
Calculate the discriminant:
\[ D = B^2 - 4AC = 25 + 4(a^2 + a - 6) \]
\[ D = 25 + 4a^2 + 4a - 24 = 4a^2 + 4a + 1 \]

### Step 6: Solve for \( x \)
Now substituting back into the quadratic formula:
\[ x = \frac{-5 \pm \sqrt{4a^2 + 4a + 1}}{2} \]
Notice that \( \sqrt{4a^2 + 4a + 1} = \sqrt{(2a + 1)^2} = |2a + 1| \).

Thus, we have:
\[ x = \frac{-5 \pm |2a + 1|}{2} \]

### Step 7: Find the roots
This gives us two possible solutions for \( x \):
1. \( x_1 = \frac{-5 + (2a + 1)}{2} = \frac{2a - 4}{2} = a - 2 \)
2. \( x_2 = \frac{-5 - (2a + 1)}{2} = \frac{-2a - 6}{2} = -a - 3 \)

### Final Solution
The roots of the equation \( x^2 + 5x - (a^2 + a - 6) = 0 \) are:
\[ x = a - 2 \quad \text{and} \quad x = -a - 3 \]

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