Solve The equation
`x^(2)+(2m+1)x+(2n+1)=0`

Options

A. Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement-1
B. Statement -1 is true, Statement -2 is true, Statement -2 is not a correct explanation for Statement -1
C. Statement -1 is true, Statement -2 is false
D. Statement -1 is false, Statement -2 is true

Explanation

To solve the equation \( x^2 + (2m + 1)x + (2n + 1) = 0 \), we will use the quadratic formula. The standard form of a quadratic equation is given by:

\[ ax^2 + bx + c = 0 \]

In our case:
- \( a = 1 \)
- \( b = 2m + 1 \)
- \( c = 2n + 1 \)

### Step 1: Identify the coefficients
We have:
- \( a = 1 \)
- \( b = 2m + 1 \)
- \( c = 2n + 1 \)

### Step 2: Apply the quadratic formula
The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Substitute the values of \( a \), \( b \), and \( c \)
Substituting the values into the quadratic formula:

\[ x = \frac{-(2m + 1) \pm \sqrt{(2m + 1)^2 - 4 \cdot 1 \cdot (2n + 1)}}{2 \cdot 1} \]

### Step 4: Simplify the expression
Now, we simplify the expression inside the square root:

\[ x = \frac{-(2m + 1) \pm \sqrt{(2m + 1)^2 - 4(2n + 1)}}{2} \]

Calculating \( (2m + 1)^2 \):

\[ (2m + 1)^2 = 4m^2 + 4m + 1 \]

Now substituting this back into the equation:

\[ x = \frac{-(2m + 1) \pm \sqrt{4m^2 + 4m + 1 - 8n - 4}}{2} \]

This simplifies to:

\[ x = \frac{-(2m + 1) \pm \sqrt{4m^2 + 4m - 8n - 3}}{2} \]

### Step 5: Final expression for \( x \)
Thus, the solutions for \( x \) can be expressed as:

\[ x = \frac{-(2m + 1) \pm \sqrt{4m^2 + 4m - 8n - 3}}{2} \]