`x^2+3sqrt(3)x+6`

Explanation

To factor the polynomial \( x^2 + 3\sqrt{3}x + 6 \), we will follow these steps:

### Step 1: Identify the coefficients
The given polynomial is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = 3\sqrt{3} \)
- \( c = 6 \)

### Step 2: Calculate \( ac \)
We need to calculate \( ac \):
\[ ac = 1 \times 6 = 6 \]

### Step 3: Find two numbers that add up to \( b \) and multiply to \( ac \)
We need to find two numbers \( m \) and \( n \) such that:
- \( m + n = b = 3\sqrt{3} \)
- \( mn = ac = 6 \)

### Step 4: Trial and error to find \( m \) and \( n \)
Let's try to find \( m \) and \( n \). We can test pairs of numbers that multiply to 6:
- \( 2\sqrt{3} \) and \( \sqrt{3} \)

Now, check if they add up:
\[ 2\sqrt{3} + \sqrt{3} = 3\sqrt{3} \]
This works!

### Step 5: Rewrite the polynomial
Now we can rewrite the polynomial using \( m \) and \( n \):
\[ x^2 + 2\sqrt{3}x + \sqrt{3}x + 6 \]

### Step 6: Factor by grouping
Group the terms:
\[ (x^2 + 2\sqrt{3}x) + (\sqrt{3}x + 6) \]
Factor out the common factors:
\[ x(x + 2\sqrt{3}) + \sqrt{3}(x + 2\sqrt{3}) \]

### Step 7: Factor out the common binomial
Now factor out the common binomial \( (x + 2\sqrt{3}) \):
\[ (x + 2\sqrt{3})(x + \sqrt{3}) \]

### Final Answer
Thus, the factorization of the polynomial \( x^2 + 3\sqrt{3}x + 6 \) is:
\[ (x + 2\sqrt{3})(x + \sqrt{3}) \]

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