Factorise by middle splitting{x^2} - 22x + 120 .

To factorize the quadratic expression \(x^2 - 22x + 120\) by the method of splitting the middle term, follow these steps:

1. **Identify coefficients:**
The general form of a quadratic equation is \(ax^2 + bx + c\).
Here, \(a = 1\), \(b = -22\), and \(c = 120\).

2. **Find two numbers:**
We need two numbers that:
- Multiply to \(a \cdot c = 1 \cdot 120 = 120\).
- Add up to \(b = -22\).

The two numbers are \(-10\) and \(-12\), because:
- \(-10 \times -12 = 120\).
- \(-10 + -12 = -22\).

3. **Split the middle term:**
Rewrite \(-22x\) as \(-10x - 12x\):
\(x^2 - 22x + 120 = x^2 - 10x - 12x + 120\).

4. **Group terms:**
Group the terms into two pairs:
\((x^2 - 10x) - (12x - 120)\).

5. **Factor each group:**
From the first group, factor out \(x\):
\(x(x - 10)\).
From the second group, factor out \(-12\):
\(-12(x - 10)\).

6. **Combine factors:**
The expression becomes:
\(x(x - 10) - 12(x - 10)\).
Factor out the common term \((x - 10)\):
\((x - 10)(x - 12)\).

7. **Final factorized form:**
The factorized form of \(x^2 - 22x + 120\) is:
\((x - 10)(x - 12)\).

**Verification:**
Expand \((x - 10)(x - 12)\):
\((x - 10)(x - 12) = x^2 - 12x - 10x + 120 = x^2 - 22x + 120\).

This confirms the factorization is correct.
**Answer:** The factorized form of \(x^2 - 22x + 120\) is \((x - 10)(x - 12)\).