Factorisation of `x^(2)+3sqrt(2)x+4` is

Options

A. `(x+2sqrt(2))(x+sqrt(2))`
B. `(x+2sqrt(2))(x-sqrt(2))`
C. `(x-2sqrt(2))(x+sqrt(2))`
D. `(x-2sqrt(2))(x-sqrt(2))`

Answer

To factor the expression \(x^2 + 3\sqrt{2}x + 4\), we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is \(x^2 + 3\sqrt{2}x + 4\). Here, the coefficients are:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 3\sqrt{2}\) (coefficient of \(x\))
- \(c = 4\) (constant term)

### Step 2: Split the middle term
We need to express the middle term \(3\sqrt{2}x\) in a way that allows us to factor by grouping. We can split it into two terms:
\[ 3\sqrt{2}x = \sqrt{2}x + 2\sqrt{2}x \]
Thus, we can rewrite the expression as:
\[ x^2 + \sqrt{2}x + 2\sqrt{2}x + 4 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (x^2 + \sqrt{2}x) + (2\sqrt{2}x + 4) \]

### Step 4: Factor out the common terms
From the first group \(x^2 + \sqrt{2}x\), we can factor out \(x\):
\[ x(x + \sqrt{2}) \]
From the second group \(2\sqrt{2}x + 4\), we can factor out \(2\):
\[ 2(\sqrt{2}x + 2) \]
Now we rewrite the expression:
\[ x(x + \sqrt{2}) + 2\sqrt{2}(x + \sqrt{2}) \]

### Step 5: Factor by grouping
Now we can factor out the common binomial factor \((x + \sqrt{2})\):
\[ (x + \sqrt{2})(x + 2\sqrt{2}) \]

### Final Answer
Thus, the factorization of \(x^2 + 3\sqrt{2}x + 4\) is:
\[ (x + \sqrt{2})(x + 2\sqrt{2}) \]

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