`x^2-21 x+90`

Answer

To factorize the polynomial \(x^2 - 21x + 90\), we can follow these steps:

### Step 1: Identify coefficients
The polynomial is in the form \(ax^2 + bx + c\), where:
- \(a = 1\)
- \(b = -21\)
- \(c = 90\)

### Step 2: Find two numbers that add up to \(b\) and multiply to \(ac\)
We need to find two numbers \(m\) and \(n\) such that:
- \(m + n = -21\) (which is \(b\))
- \(m \cdot n = 90\) (which is \(ac\))

### Step 3: Determine the signs of \(m\) and \(n\)
Since \(m + n\) is negative and \(m \cdot n\) is positive, both \(m\) and \(n\) must be negative numbers.

### Step 4: Find suitable pairs of factors of 90
The pairs of factors of 90 are:
- (-1, -90)
- (-2, -45)
- (-3, -30)
- (-5, -18)
- (-6, -15)
- (-9, -10)

We need to find a pair that adds up to -21.

### Step 5: Identify the correct pair
The pair that works is:
- \(m = -15\)
- \(n = -6\)

This is because:
- \(-15 + (-6) = -21\)
- \(-15 \cdot -6 = 90\)

### Step 6: Rewrite the polynomial
Now we can rewrite the polynomial using \(m\) and \(n\):
\[ x^2 - 15x - 6x + 90 \]

### Step 7: Group the terms
Next, we group the terms:
\[ (x^2 - 15x) + (-6x + 90) \]

### Step 8: Factor by grouping
Now we factor out the common factors in each group:
\[ x(x - 15) - 6(x - 15) \]

### Step 9: Factor out the common binomial
Now we can factor out the common binomial \((x - 15)\):
\[ (x - 15)(x - 6) \]

### Final Answer
Thus, the factorization of the polynomial \(x^2 - 21x + 90\) is:
\[ (x - 15)(x - 6) \]

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