factorize the given expression `sqrt(2)x^2+3x+sqrt(2)`

Explanation

To factorize the expression \( \sqrt{2}x^2 + 3x + \sqrt{2} \), we will use the middle term splitting method. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given expression is \( \sqrt{2}x^2 + 3x + \sqrt{2} \).
- Coefficient of \( x^2 \) (a) = \( \sqrt{2} \)
- Coefficient of \( x \) (b) = \( 3 \)
- Constant term (c) = \( \sqrt{2} \)

### Step 2: Multiply the coefficients
We need to multiply the coefficient of \( x^2 \) (which is \( \sqrt{2} \)) and the constant term (which is also \( \sqrt{2} \)):
\[ a \cdot c = \sqrt{2} \cdot \sqrt{2} = 2 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( 2 \) (the result from step 2) and add up to \( 3 \) (the coefficient of \( x \)):
- The factors of \( 2 \) that add up to \( 3 \) are \( 2 \) and \( 1 \).

### Step 4: Rewrite the middle term
Now we can rewrite the expression by splitting the middle term \( 3x \) into \( 2x + 1x \):
\[ \sqrt{2}x^2 + 2x + 1x + \sqrt{2} \]

### Step 5: Group the terms
Next, we group the terms:
\[ (\sqrt{2}x^2 + 2x) + (1x + \sqrt{2}) \]

### Step 6: Factor out the common terms in each group
Now we factor out the common terms from each group:
- From the first group \( \sqrt{2}x^2 + 2x \), we can factor out \( 2x \):
\[ 2x(\frac{\sqrt{2}}{2}x + 1) \]
- From the second group \( 1x + \sqrt{2} \), we can factor out \( 1 \):
\[ 1(x + \sqrt{2}) \]

### Step 7: Combine the factors
Now we can combine the factors:
\[ (\sqrt{2}x + 1)(x + \sqrt{2}) \]

### Final Answer
Thus, the factorized form of the expression \( \sqrt{2}x^2 + 3x + \sqrt{2} \) is:
\[ (\sqrt{2}x + 1)(x + \sqrt{2}) \]

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