Here's the detailed breakdown and solutions for the given questions:

1. Simplify $\sqrt{-25} \cdot \sqrt{-81}$:

• $\sqrt{-25} = 5i$ (since $\sqrt{-1} = i$),
• $\sqrt{-81} = 9i$,
• Multiplying them: $5i \cdot 9i = 45i^2$,
• Since $i^2 = -1$, we have $45i^2 = 45 \cdot (-1) = -45$.

Answer: $-45$.

2. Solve $x^2 + 16 = 0$:

• Rewrite the equation: $x^2 = -16$,
• Taking square roots: $x = \pm\sqrt{-16}$,
• Since $\sqrt{-16} = 4i$, we get $x = \pm 4i$.

Answer: $x = 4i$ and $x = -4i$ (Options a and b).

3. Multiply $(3 + 5i)(4 + 2i)$:

Using the distributive property:

• First: $3 \cdot 4 = 12$,
• Outer: $3 \cdot 2i = 6i$,
• Inner: $5i \cdot 4 = 20i$,
• Last: $5i \cdot 2i = 10i^2$.

Combine terms:

• $12 + 6i + 20i + 10i^2$,
• Since $i^2 = -1$, $10i^2 = -10$,
• Combine real and imaginary parts: $12 - 10 + (6i + 20i) = 2 + 26i$.

Answer: $2 + 26i$.

Would you like further clarification on any of these solutions or the steps involved?

Related Questions:

1. What are the properties of imaginary numbers like $i^2, i^3$, and $i^4$?
2. How do you simplify square roots of negative numbers in general?
3. What is the general method for solving quadratic equations with complex roots?
4. How do you multiply complex numbers geometrically (using polar form)?
5. Can you add and subtract complex numbers in a similar way to multiplying them?

Tip: Always check the signs carefully when working with $i^2$, as it changes the nature of the solution.