`6 sqrt5 x ^(2) - 2x - 4 sqrt5 ` is equal to

Options

A. `(sqrt5 x -2) ( 6x + 2 sqrt5)`
B. `( sqrt5 x + 2) ( 6x + 2sqrt5)`
C. `(sqrt5 x + 2) ( 6x -2 sqrt5)`
D. `(sqrt5) x -2) ( 6x -2 sqrt5)`

Answer

To factor the expression \( 6\sqrt{5}x^2 - 2x - 4\sqrt{5} \), we can follow these steps:

### Step 1: Identify the coefficients
The expression can be rewritten in the standard quadratic form \( ax^2 + bx + c \):
- \( a = 6\sqrt{5} \)
- \( b = -2 \)
- \( c = -4\sqrt{5} \)

### Step 2: Calculate the product \( ac \)
We need to find the product of \( a \) and \( c \):
\[ ac = (6\sqrt{5})(-4\sqrt{5}) = -24 \cdot 5 = -120 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need two numbers that multiply to \(-120\) and add to \(-2\). The numbers that satisfy this condition are \(10\) and \(-12\) since:
\[ 10 \times (-12) = -120 \quad \text{and} \quad 10 + (-12) = -2 \]

### Step 4: Rewrite the middle term
Using the numbers found, we can rewrite the expression:
\[ 6\sqrt{5}x^2 - 12x + 10x - 4\sqrt{5} \]

### Step 5: Group the terms
Now, we can group the terms:
\[ (6\sqrt{5}x^2 - 12x) + (10x - 4\sqrt{5}) \]

### Step 6: Factor out the common terms from each group
From the first group \( (6\sqrt{5}x^2 - 12x) \), we can factor out \( 6x \):
\[ 6x(\sqrt{5}x - 2) \]
From the second group \( (10x - 4\sqrt{5}) \), we can factor out \( 2 \):
\[ 2(5x - 2\sqrt{5}) \]

### Step 7: Combine the factored terms
Now we can write the expression as:
\[ 6x(\sqrt{5}x - 2) + 2(5x - 2\sqrt{5}) \]

### Step 8: Factor out the common binomial
Notice that \( (\sqrt{5}x - 2) \) is common in both terms:
\[ (6x + 2\sqrt{5})(\sqrt{5}x - 2) \]

### Final Factored Form
Thus, the factored form of the expression \( 6\sqrt{5}x^2 - 2x - 4\sqrt{5} \) is:
\[ (6x + 2\sqrt{5})(\sqrt{5}x - 2) \]