Factor the following expressions.
`x^(2)-5x-66`

Answer

To factor the quadratic expression \( x^2 - 5x - 66 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -5 \)
- \( c = -66 \)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \cdot (-66) = -66 \) and add to \( b = -5 \).

The two numbers that satisfy these conditions are \( -11 \) and \( 6 \) because:
- \( -11 \times 6 = -66 \)
- \( -11 + 6 = -5 \)

### Step 3: Rewrite the middle term
Now, we can rewrite the expression \( x^2 - 5x - 66 \) using the two numbers we found:
\[ x^2 - 11x + 6x - 66 \]

### Step 4: Factor by grouping
Next, we will group the terms:
\[ (x^2 - 11x) + (6x - 66) \]

Now, we factor out the common factors from each group:
- From the first group \( x^2 - 11x \), we can factor out \( x \):
\[ x(x - 11) \]
- From the second group \( 6x - 66 \), we can factor out \( 6 \):
\[ 6(x - 11) \]

### Step 5: Combine the factors
Now we can combine the factored groups:
\[ x(x - 11) + 6(x - 11) = (x - 11)(x + 6) \]

### Final Answer
Thus, the factored form of the expression \( x^2 - 5x - 66 \) is:
\[ (x - 11)(x + 6) \]

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