One of the factors of the expression
` 4 sqrt(3) x^(2) + 5 x - 2 sqrt(3) ` is

Options

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

Answer

To find one of the factors of the expression \( 4\sqrt{3}x^2 + 5x - 2\sqrt{3} \), we can follow these steps:

### Step 1: Identify the expression and set it to zero
We start with the expression:
\[ 4\sqrt{3}x^2 + 5x - 2\sqrt{3} = 0 \]

### Step 2: Multiply the leading coefficient and the constant term
Next, we multiply the coefficient of \(x^2\) (which is \(4\sqrt{3}\)) by the constant term (which is \(-2\sqrt{3}\)):
\[ 4\sqrt{3} \times -2\sqrt{3} = -8 \times 3 = -24 \]

### Step 3: Find two numbers that multiply to give -24 and add to give 5
We need to find two numbers that multiply to \(-24\) and add up to \(5\). The numbers that satisfy this condition are \(8\) and \(-3\):
\[ 8 \times -3 = -24 \quad \text{and} \quad 8 + (-3) = 5 \]

### Step 4: Rewrite the expression using these two numbers
We can rewrite the middle term \(5x\) using \(8x\) and \(-3x\):
\[ 4\sqrt{3}x^2 + 8x - 3x - 2\sqrt{3} = 0 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (4\sqrt{3}x^2 + 8x) + (-3x - 2\sqrt{3}) = 0 \]

### Step 6: Factor by grouping
Next, we factor out the common factors from each group:
\[ 4x(\sqrt{3}x + 2) - \sqrt{3}(3x + 2) = 0 \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \((\sqrt{3}x + 2)\):
\[ (\sqrt{3}x + 2)(4x - \sqrt{3}) = 0 \]

### Step 8: Identify the factors
From the factored form, we can see that the factors of the expression are:
\[ \sqrt{3}x + 2 \quad \text{and} \quad 4x - \sqrt{3} \]

### Conclusion
Thus, one of the factors of the expression \(4\sqrt{3}x^2 + 5x - 2\sqrt{3}\) is:
\[ 4x - \sqrt{3} \]