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

Options

A. `(x+3sqrt(3))(sqrt(3)x-1)`
B. `(x-3sqrt(3))(sqrt(3)x-1)`
C. `(x+3sqrt(3))(sqrt(3)x+1)`
D. `(x-3sqrt(3))(sqrt(3)x+1)`

Explanation

To factorise the expression \( \sqrt{3}x^2 + 10x + 3\sqrt{3} \), we can follow these steps:

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

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ a \cdot c = \sqrt{3} \cdot 3\sqrt{3} = 3 \cdot 3 = 9 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( 9 \) and add up to \( 10 \). The numbers are \( 9 \) and \( 1 \) because:
\[ 9 \cdot 1 = 9 \quad \text{and} \quad 9 + 1 = 10 \]

### Step 4: Rewrite the middle term
Now we can rewrite the polynomial by splitting the middle term using the two numbers found:
\[ \sqrt{3}x^2 + 9x + 1x + 3\sqrt{3} \]

### Step 5: Group the terms
Next, we group the terms:
\[ (\sqrt{3}x^2 + 9x) + (1x + 3\sqrt{3}) \]

### Step 6: Factor by grouping
Now, we can factor out the common factors from each group:
\[ x(\sqrt{3}x + 9) + 1(\sqrt{3}x + 3) \]

### Step 7: Factor out the common binomial
Notice that \( \sqrt{3}x + 9 \) and \( \sqrt{3}x + 3 \) share a common factor:
\[ (\sqrt{3}x + 3)(x + 1) \]

### Final Answer
Thus, the factorization of \( \sqrt{3}x^2 + 10x + 3\sqrt{3} \) is:
\[ (\sqrt{3}x + 3)(x + 1) \]

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