Form a quadratic equation whose roots are `2sqrt(3)-5 and -2sqrt(3)-5.`

Answer

To form a quadratic equation whose roots are \(2\sqrt{3} - 5\) and \(-2\sqrt{3} - 5\), we will follow these steps:

### Step 1: Identify the Roots
Let the roots be:
- \(\alpha = 2\sqrt{3} - 5\)
- \(\beta = -2\sqrt{3} - 5\)

### Step 2: Calculate the Sum of the Roots
The sum of the roots \((\alpha + \beta)\) can be calculated as follows:
\[ \alpha + \beta = (2\sqrt{3} - 5) + (-2\sqrt{3} - 5) \]
Simplifying this:
\[ \alpha + \beta = 2\sqrt{3} - 5 - 2\sqrt{3} - 5 = -10 \]

### Step 3: Calculate the Product of the Roots
The product of the roots \((\alpha \cdot \beta)\) can be calculated as:
\[ \alpha \cdot \beta = (2\sqrt{3} - 5)(-2\sqrt{3} - 5) \]
Using the difference of squares:
\[ \alpha \cdot \beta = -( (2\sqrt{3})^2 - 5^2 ) \]
Calculating the squares:
\[ = - (12 - 25) = -(-13) = 13 \]

### Step 4: Form the Quadratic Equation
The standard form of a quadratic equation with roots \(\alpha\) and \(\beta\) is given by:
\[ x^2 - (\alpha + \beta)x + (\alpha \cdot \beta) = 0 \]
Substituting the values we found:
\[ x^2 - (-10)x + 13 = 0 \]
This simplifies to:
\[ x^2 + 10x + 13 = 0 \]

### Final Result
The required quadratic equation is:
\[ \boxed{x^2 + 10x + 13 = 0} \]