The positive square root of `(2x ^(2) + 5x + 2) (x ^(2) - 4) ( 2x ^(2) - 3x -2)` expressed as factor is:

Options

A. `(2x + 3) (x+2) (x -1)`
B. `(x +2) ( x-2) (2x -1)`
C. `(2x +1)(x +2) (x-2)`
D. `(2x +1) (x -2) ( x-1)`

Answer

To find the positive square root of the expression \((2x^2 + 5x + 2)(x^2 - 4)(2x^2 - 3x - 2)\) expressed as factors, we will follow these steps:

### Step 1: Factor each component of the expression

1. **Factor \(2x^2 + 5x + 2\)**:
- We need to split the middle term. We look for two numbers that multiply to \(2 \times 2 = 4\) and add up to \(5\). The numbers \(1\) and \(4\) work.
- Rewrite: \(2x^2 + 1x + 4x + 2\).
- Group: \((2x^2 + 1x) + (4x + 2)\).
- Factor: \(x(2x + 1) + 2(2x + 1)\).
- Combine: \((2x + 1)(x + 2)\).

2. **Factor \(x^2 - 4\)**:
- This is a difference of squares: \(x^2 - 2^2\).
- Factor: \((x - 2)(x + 2)\).

3. **Factor \(2x^2 - 3x - 2\)**:
- We need to split the middle term. We look for two numbers that multiply to \(2 \times -2 = -4\) and add up to \(-3\). The numbers \(-4\) and \(1\) work.
- Rewrite: \(2x^2 - 4x + x - 2\).
- Group: \((2x^2 - 4x) + (x - 2)\).
- Factor: \(2x(x - 2) + 1(x - 2)\).
- Combine: \((2x + 1)(x - 2)\).

### Step 2: Combine all factors

Now we can combine all the factors:
\[ (2x + 1)(x + 2)(x - 2)(x - 2)(2x + 1) \]

### Step 3: Simplify the expression

Notice that we have:
- \((2x + 1)^2\)
- \((x + 2)^2\)
- \((x - 2)^2\)

So the expression can be rewritten as:
\[ (2x + 1)^2 (x + 2)^2 (x - 2)^2 \]

### Step 4: Find the positive square root

Taking the positive square root of the entire expression gives:
\[ (2x + 1)(x + 2)(x - 2) \]

### Final Answer

Thus, the positive square root of the expression \((2x^2 + 5x + 2)(x^2 - 4)(2x^2 - 3x - 2)\) expressed as factors is:
\[ (2x + 1)(x + 2)(x - 2) \]

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