`sqrt(3)x^2+10x+8sqrt(3)`

Answer

To factor the polynomial \( \sqrt{3}x^2 + 10x + 8\sqrt{3} \), we will follow these steps:

### Step 1: Identify the coefficients
The polynomial is in the form \( ax^2 + bx + c \), where:
- \( a = \sqrt{3} \)
- \( b = 10 \)
- \( c = 8\sqrt{3} \)

### Step 2: Calculate \( ac \)
We need to find the product \( ac \):
\[ ac = \sqrt{3} \times 8\sqrt{3} = 8 \times 3 = 24 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( 24 \) (the value of \( ac \)) and add up to \( 10 \) (the value of \( b \)). The numbers are \( 6 \) and \( 4 \) because:
\[ 6 \times 4 = 24 \quad \text{and} \quad 6 + 4 = 10 \]

### Step 4: Rewrite the middle term
We can rewrite the polynomial by splitting the middle term using the numbers we found:
\[ \sqrt{3}x^2 + 6x + 4x + 8\sqrt{3} \]

### Step 5: Group the terms
Now, we will group the terms:
\[ (\sqrt{3}x^2 + 6x) + (4x + 8\sqrt{3}) \]

### Step 6: Factor out the common terms in each group
In the first group \( \sqrt{3}x^2 + 6x \), we can factor out \( 2\sqrt{3}x \):
\[ 2\sqrt{3}x(x + 2) \]
In the second group \( 4x + 8\sqrt{3} \), we can factor out \( 4 \):
\[ 4(x + 2\sqrt{3}) \]

### Step 7: Combine the factors
Now we can combine the factored groups:
\[ 2\sqrt{3}x(x + 2) + 4(x + 2\sqrt{3}) \]
This can be rewritten as:
\[ (x + 2\sqrt{3})(\sqrt{3}x + 4) \]

### Final Factorization
Thus, the factors of the polynomial \( \sqrt{3}x^2 + 10x + 8\sqrt{3} \) are:
\[ (x + 2\sqrt{3})(\sqrt{3}x + 4) \]

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