The twice of the product of real roots of the equation ` (2x+3)^(2)- 3|2x+3|+2=0` is __________

Answer

To solve the equation \( (2x + 3)^2 - 3|2x + 3| + 2 = 0 \) and find twice the product of its real roots, we will follow these steps:

### Step 1: Substitute for simplification
Let \( y = 2x + 3 \). Then, we can rewrite the equation as:
\[ y^2 - 3|y| + 2 = 0 \]

### Step 2: Consider cases for the absolute value
We need to consider two cases for \( |y| \).

**Case 1:** \( y \geq 0 \) (which means \( |y| = y \))
The equation becomes:
\[ y^2 - 3y + 2 = 0 \]
Factoring this, we have:
\[ (y - 1)(y - 2) = 0 \]
Thus, the roots are:
\[ y = 1 \quad \text{and} \quad y = 2 \]

**Case 2:** \( y < 0 \) (which means \( |y| = -y \))
The equation becomes:
\[ y^2 + 3y + 2 = 0 \]
Factoring this, we have:
\[ (y + 1)(y + 2) = 0 \]
Thus, the roots are:
\[ y = -1 \quad \text{and} \quad y = -2 \]

### Step 3: Find corresponding values of \( x \)
Now we will convert the values of \( y \) back to \( x \) using \( y = 2x + 3 \).

1. For \( y = 1 \):
\[ 1 = 2x + 3 \implies 2x = 1 - 3 \implies 2x = -2 \implies x = -1 \]

2. For \( y = 2 \):
\[ 2 = 2x + 3 \implies 2x = 2 - 3 \implies 2x = -1 \implies x = -\frac{1}{2} \]

3. For \( y = -1 \):
\[ -1 = 2x + 3 \implies 2x = -1 - 3 \implies 2x = -4 \implies x = -2 \]

4. For \( y = -2 \):
\[ -2 = 2x + 3 \implies 2x = -2 - 3 \implies 2x = -5 \implies x = -\frac{5}{2} \]

### Step 4: List the real roots
The real roots of the original equation are:
\[ x = -1, \quad x = -\frac{1}{2}, \quad x = -2, \quad x = -\frac{5}{2} \]

### Step 5: Calculate the product of the real roots
The product of the real roots is:
\[ (-1) \times \left(-\frac{1}{2}\right) \times (-2) \times \left(-\frac{5}{2}\right) \]
Calculating this step-by-step:
1. \( (-1) \times (-2) = 2 \)
2. \( \left(-\frac{1}{2}\right) \times \left(-\frac{5}{2}\right) = \frac{5}{4} \)
3. Now, multiply the results:
\[ 2 \times \frac{5}{4} = \frac{10}{4} = \frac{5}{2} \]

### Step 6: Find twice the product
Twice the product of the real roots is:
\[ 2 \times \frac{5}{2} = 5 \]

### Final Answer
The twice of the product of the real roots is **5**.

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