Foctorise :
` x^(2) - 2x- 9 `

Answer

To factorize the expression \( x^2 - 2x - 9 \), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ x^2 - 2x - 9 \]

### Step 2: Identify a Perfect Square
We notice that the first two terms, \( x^2 - 2x \), can be rearranged to form a perfect square. We can express \( -2x \) as \( -2 \cdot x \cdot 1 \). To complete the square, we take half of the coefficient of \( x \) (which is -2), square it, and add and subtract this value:
\[ x^2 - 2x + 1 - 1 - 9 \]
This simplifies to:
\[ (x - 1)^2 - 10 \]

### Step 3: Recognize the Difference of Squares
Now we have:
\[ (x - 1)^2 - 10 \]
We can recognize this as a difference of squares, where \( a = (x - 1) \) and \( b = \sqrt{10} \):
\[ a^2 - b^2 = (a + b)(a - b) \]
Thus, we can factor it as:
\[ ((x - 1) + \sqrt{10})((x - 1) - \sqrt{10}) \]

### Step 4: Write the Final Factored Form
The final factored form of the expression \( x^2 - 2x - 9 \) is:
\[ (x - 1 + \sqrt{10})(x - 1 - \sqrt{10}) \]

### Summary
The expression \( x^2 - 2x - 9 \) can be factorized as:
\[ (x - 1 + \sqrt{10})(x - 1 - \sqrt{10}) \]