To simplify the given expression to the form $a + bi$, let's break down each part:

Expression:

$-\sqrt{36} + \sqrt{-128} - \sqrt{36} + \sqrt{-200}$

Step 1: Simplify each square root.

1. $\sqrt{36} = 6$, so $-\sqrt{36} = -6$.
2. $\sqrt{-128}$ can be rewritten as $\sqrt{128} \cdot i$.
   • $\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}$, so $\sqrt{-128} = 8\sqrt{2} \cdot i$.
3. $\sqrt{36} = 6$, so $-\sqrt{36} = -6$.
4. $\sqrt{-200}$ can be rewritten as $\sqrt{200} \cdot i$.
   • $\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}$, so $\sqrt{-200} = 10\sqrt{2} \cdot i$.

Step 2: Substitute these simplified values back into the expression.

$-6 + 8\sqrt{2}i - 6 + 10\sqrt{2}i$

Step 3: Combine like terms.

• Combine the real parts: $-6 - 6 = -12$.
• Combine the imaginary parts: $8\sqrt{2}i + 10\sqrt{2}i = 18\sqrt{2}i$.

Final Answer:

The simplified expression in the form $a + bi$ is: $-12 + 18\sqrt{2}i$

Would you like any more details or have any questions?

Here are some related questions you might find interesting:

1. How do you simplify expressions involving complex numbers?
2. What are the rules for combining real and imaginary parts in complex numbers?
3. How can you find the magnitude of a complex number?
4. How can you graph a complex number on the complex plane?
5. What is the significance of the imaginary unit $i$ in mathematics?

Tip: When working with square roots of negative numbers, always express them in terms of the imaginary unit $i$.