Factorise: `x ^(2) + 5 sqrt3 x + 12 `

Options

A. `(x + 2 sqrt3 ) (x + 3 sqrt3)`
B. `(x + 2 sqrt3 ) (x + 4 sqrt3)`
C. `(x + 4 sqrt3) (x - sqrt3)`
D. `(x - sqrt3) (x -4 sqrt3)`

Answer

To factor the expression \( x^2 + 5\sqrt{3}x + 12 \), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the standard quadratic form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = 5\sqrt{3} \)
- \( c = 12 \)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \cdot 12 = 12 \) and add up to \( b = 5\sqrt{3} \).

### Step 3: Determine the two numbers
We can express \( 5\sqrt{3} \) as a sum of two terms that can be factored. We can try \( 4\sqrt{3} \) and \( \sqrt{3} \):
- \( 4\sqrt{3} + \sqrt{3} = 5\sqrt{3} \)
- \( 4\sqrt{3} \cdot \sqrt{3} = 4 \cdot 3 = 12 \)

### Step 4: Rewrite the expression
Now we can rewrite the quadratic expression using these two numbers:
\[ x^2 + 4\sqrt{3}x + \sqrt{3}x + 12 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (x^2 + 4\sqrt{3}x) + (\sqrt{3}x + 12) \]

### Step 6: Factor by grouping
Now, we factor out the common terms in each group:
\[ x(x + 4\sqrt{3}) + \sqrt{3}(x + 4\sqrt{3}) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (x + 4\sqrt{3}) \):
\[ (x + \sqrt{3})(x + 4\sqrt{3}) \]

### Final Answer
Thus, the factorized form of the expression \( x^2 + 5\sqrt{3}x + 12 \) is:
\[ (x + \sqrt{3})(x + 4\sqrt{3}) \]

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