If `p, q, r, s in R`, then equaton `(x^2 + px + 3q) (-x^2 + rx + q) (-x^2 + sx-2q) = 0` has

Options

A. 6 real roots
B. at least two real roots
C. 2 real and 4 imaginary roots
D. 4 real and 2 imaginary roots

Answer

To solve the equation \((x^2 + px + 3q)(-x^2 + rx + q)(-x^2 + sx - 2q) = 0\), we need to analyze the nature of the roots of each quadratic factor.

### Step 1: Identify the quadratic equations
The equation consists of three quadratic factors:
1. \(f_1(x) = x^2 + px + 3q\)
2. \(f_2(x) = -x^2 + rx + q\)
3. \(f_3(x) = -x^2 + sx - 2q\)

### Step 2: Calculate the discriminant of each quadratic
The discriminant \(D\) of a quadratic \(ax^2 + bx + c\) is given by \(D = b^2 - 4ac\).

1. For \(f_1(x)\):
\[ D_1 = p^2 - 4 \cdot 1 \cdot 3q = p^2 - 12q \]

2. For \(f_2(x)\):
\[ D_2 = r^2 - 4 \cdot (-1) \cdot q = r^2 + 4q \]

3. For \(f_3(x)\):
\[ D_3 = s^2 - 4 \cdot (-1) \cdot (-2q) = s^2 - 8q \]

### Step 3: Analyze the discriminants based on the value of \(q\)
We will analyze the nature of roots based on whether \(q\) is greater than or less than zero.

#### Case 1: \(q > 0\)
- \(D_1 = p^2 - 12q\): This can be positive or negative depending on the value of \(p\).
- \(D_2 = r^2 + 4q\): This is always positive since \(4q > 0\).
- \(D_3 = s^2 - 8q\): This can also be positive or negative depending on the value of \(s\).

In this case, we can conclude:
- At least one of \(D_1\) or \(D_3\) could be negative, leading to potentially 2 real roots and 4 imaginary roots.

#### Case 2: \(q < 0\)
- \(D_1 = p^2 - 12q\): This is always positive since \(-12q > 0\).
- \(D_2 = r^2 + 4q\): This can be positive or negative depending on \(r\).
- \(D_3 = s^2 - 8q\): This is always positive since \(-8q > 0\).

In this case, we can conclude:
- At least 2 real roots from \(D_1\) and potentially 2 more from \(D_2\) depending on \(r\), leading to a maximum of 4 real roots.

### Conclusion
From the analysis:
- If \(q > 0\), we can have at least 2 real roots and potentially 4 imaginary roots.
- If \(q < 0\), we can have at least 2 real roots and potentially 4 real roots.

Thus, the equation has at least 2 real roots in both cases.

### Final Answer
The correct option is: **At least two real roots.**