If `(-2 + sqrt-3) (-3 + 2 sqrt-3)=a +bi`, find the real numbers a and b with values of a and b, also find the modulus of `a+bi`

Answer

To solve the problem, we need to evaluate the expression \((-2 + \sqrt{-3})(-3 + 2\sqrt{-3})\) and express it in the form \(a + bi\), where \(a\) and \(b\) are real numbers. After that, we will find the modulus of the complex number.

### Step-by-Step Solution:

1. **Rewrite the Complex Numbers**:
We start by rewriting the square roots of negative numbers using \(i\), where \(i = \sqrt{-1}\). Thus:
\[ \sqrt{-3} = i\sqrt{3} \]
Therefore, we can rewrite the expression as:
\[ (-2 + i\sqrt{3})(-3 + 2i\sqrt{3}) \]

2. **Expand the Expression**:
We will now use the distributive property (FOIL method) to expand the expression:
\[ (-2)(-3) + (-2)(2i\sqrt{3}) + (i\sqrt{3})(-3) + (i\sqrt{3})(2i\sqrt{3}) \]
This simplifies to:
\[ 6 - 4i\sqrt{3} - 3i\sqrt{3} + 2(i^2)(\sqrt{3})(\sqrt{3}) \]
Since \(i^2 = -1\), we have:
\[ 2(-1)(\sqrt{3})(\sqrt{3}) = -6 \]

3. **Combine Like Terms**:
Now we can combine all the terms:
\[ 6 - 6 - 4i\sqrt{3} - 3i\sqrt{3} = 0 - 7i\sqrt{3} \]
This gives us:
\[ 0 - 7i\sqrt{3} = 0 + (-7\sqrt{3})i \]

4. **Identify Real and Imaginary Parts**:
From the expression \(0 - 7i\sqrt{3}\), we can identify:
\[ a = 0 \quad \text{and} \quad b = -7\sqrt{3} \]

5. **Find the Modulus**:
The modulus of a complex number \(a + bi\) is given by:
\[ |a + bi| = \sqrt{a^2 + b^2} \]
Substituting the values of \(a\) and \(b\):
\[ |0 + (-7\sqrt{3})i| = \sqrt{0^2 + (-7\sqrt{3})^2} = \sqrt{0 + 49 \cdot 3} = \sqrt{147} \]

### Final Answers:
- The values of \(a\) and \(b\) are:
\[ a = 0, \quad b = -7\sqrt{3} \]
- The modulus of \(a + bi\) is:
\[ |a + bi| = \sqrt{147} \]