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

Answer

To solve the quadratic equation \( x^2 + 6x + 5 = 0 \), we will factorize it step by step.

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 5 \)

### Step 2: Factor the quadratic expression
We need to find two numbers that multiply to \( ac = 1 \times 5 = 5 \) and add up to \( b = 6 \).

The two numbers that satisfy these conditions are \( 1 \) and \( 5 \) because:
- \( 1 \times 5 = 5 \)
- \( 1 + 5 = 6 \)

### Step 3: Rewrite the quadratic equation
Using the numbers found, we can rewrite the middle term:
\[ x^2 + 1x + 5x + 5 = 0 \]

### Step 4: Group the terms
Now, we can group the terms:
\[ (x^2 + 1x) + (5x + 5) = 0 \]

### Step 5: Factor by grouping
Now we factor out the common terms from each group:
\[ x(x + 1) + 5(x + 1) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (x + 1) \):
\[ (x + 1)(x + 5) = 0 \]

### Step 7: Set each factor to zero
Now we set each factor equal to zero:
1. \( x + 1 = 0 \)
2. \( x + 5 = 0 \)

### Step 8: Solve for \( x \)
From the first equation:
\[ x + 1 = 0 \implies x = -1 \]
From the second equation:
\[ x + 5 = 0 \implies x = -5 \]

### Conclusion
The roots of the quadratic equation \( x^2 + 6x + 5 = 0 \) are:
\[ x = -1 \quad \text{and} \quad x = -5 \]

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