To solve the equation \(4b^2 + 320 = 0\), follow these steps:

1. **Isolate the quadratic term**:
   \[
   4b^2 = -320
   \]

2. **Divide both sides by 4**:
   \[
   b^2 = -80
   \]

3. **Take the square root of both sides**:
   \[
   b = \pm \sqrt{-80}
   \]

4. **Simplify the square root**:
   Since \(\sqrt{-80} = \sqrt{-1 \times 16 \times 5} = \sqrt{-1} \times \sqrt{16} \times \sqrt{5} = 4i\sqrt{5}\), where \(i\) is the imaginary unit, we have:
   \[
   b = \pm 4i\sqrt{5}
   \]

So, the solutions are:
\[
b = 4i\sqrt{5}, \quad b = -4i\sqrt{5}
\]

Would you like more details or have any questions?

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Here are five related questions to explore further:

1. What is the general method for solving quadratic equations?
2. How do imaginary numbers come into play when solving quadratic equations?
3. What are the properties of the imaginary unit \(i\)?
4. How would you factorize a quadratic equation involving complex numbers?
5. How can the quadratic formula be applied to solve quadratic equations?

**Tip:** When dealing with negative numbers under the square root, remember that the square root of \(-1\) is represented by the imaginary unit \(i\).

Learn how to solve the quadratic equation 4b^2 + 320 = 0 with complex roots involving imaginary numbers. This problem covers key algebraic concepts, including the quadratic equation formula and handling complex roots. Suitable for advanced students in Grades 10-12.

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