`x^2-32x-105`

Answer

To factor the quadratic polynomial \( x^2 - 32x - 105 \), we can use the method of middle term splitting. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given polynomial is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -32 \)
- \( c = -105 \)

### Step 2: Find the product and sum
We need to find two numbers that:
- Multiply to \( ac = 1 \times (-105) = -105 \)
- Add up to \( b = -32 \)

### Step 3: List the factor pairs of -105
The factor pairs of -105 are:
- \( 1 \) and \( -105 \)
- \( -1 \) and \( 105 \)
- \( 3 \) and \( -35 \)
- \( -3 \) and \( 35 \)
- \( 5 \) and \( -21 \)
- \( -5 \) and \( 21 \)
- \( 7 \) and \( -15 \)
- \( -7 \) and \( 15 \)

### Step 4: Identify the correct pair
From the factor pairs, we need to find a pair that adds up to -32. The correct pair is:
- \( -35 \) and \( 3 \) because \( -35 + 3 = -32 \)

### Step 5: Rewrite the middle term
We can rewrite the polynomial using the numbers we found:
\[ x^2 - 35x + 3x - 105 \]

### Step 6: Group the terms
Now, we group the terms:
\[ (x^2 - 35x) + (3x - 105) \]

### Step 7: Factor by grouping
Now, we factor out the common factors from each group:
\[ x(x - 35) + 3(x - 35) \]

### Step 8: Factor out the common binomial
Now, we can factor out the common binomial \( (x - 35) \):
\[ (x - 35)(x + 3) \]

### Final Answer
Thus, the factorization of the polynomial \( x^2 - 32x - 105 \) is:
\[ (x - 35)(x + 3) \]