`x^2-x-156`

Answer

To factor the polynomial \(x^2 - x - 156\), we will follow these steps:

### Step 1: Identify the polynomial
We start with the quadratic polynomial:
\[ x^2 - x - 156 \]

### Step 2: Find two numbers that multiply to -156 and add to -1
We need to find two numbers \(a\) and \(b\) such that:
\[ a \times b = -156 \quad \text{and} \quad a + b = -1 \]
To do this, we can list the factors of 156 and check their combinations.

### Step 3: List the factors of 156
The factors of 156 are:
- \(1 \times 156\)
- \(2 \times 78\)
- \(3 \times 52\)
- \(4 \times 39\)
- \(6 \times 26\)
- \(12 \times 13\)

### Step 4: Find the correct pair
From the factors, we can check pairs:
- \(12\) and \(-13\) multiply to \(-156\) and add to \(-1\):
\[ 12 + (-13) = -1 \]

### Step 5: Rewrite the polynomial
Now we can rewrite the polynomial using the numbers we found:
\[ x^2 - 13x + 12x - 156 \]

### Step 6: Group the terms
Next, we group the terms:
\[ (x^2 - 13x) + (12x - 156) \]

### Step 7: Factor by grouping
Now we factor out the common factors from each group:
\[ x(x - 13) + 12(x - 13) \]

### Step 8: Factor out the common binomial
Now we can factor out the common binomial \((x - 13)\):
\[ (x - 13)(x + 12) \]

### Final Answer
Thus, the factorization of the polynomial \(x^2 - x - 156\) is:
\[ (x - 13)(x + 12) \]

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