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Let's go step-by-step to solve each question in your quiz:

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### **1. Evaluate the following:**  
**(a)** \((2 - 4i) - (-4 + 3i)\)  
Distribute the negative sign:  
\[(2 - 4i) + 4 - 3i\]  
Now combine like terms:  
\[2 + 4 = 6\]  
\[-4i - 3i = -7i\]  
The result is:  
\[\boxed{6 - 7i}\]  

**(b)** \(i^8\)  
Recall that powers of \(i\) follow a cycle:  
\[
i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and the cycle repeats.}
\]  
Since \(8 \div 4 = 2\) with no remainder, \(i^8\) corresponds to \(i^4 = 1\).  
Thus:  
\[\boxed{1}\]

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### **2. Let \(f(x) = x^2 (x - 3)^2 (x - 5) (x + 3)\). Find the solution set (in interval notation) for \(f(x) < 0\).**

1. Identify the roots:
   - \(x = 0, 3, 5, -3\).

2. Analyze the sign changes around these points.
   - Between the roots, the factors \(x^2\), \((x - 3)^2\), \((x - 5)\), and \((x + 3)\) may switch sign.

3. Since \(x^2\) and \((x - 3)^2\) are always non-negative, the sign changes only depend on \((x - 5)\) and \((x + 3)\).

4. Test intervals between the roots:
   - \(x < -3\): All factors negative, so \(f(x) > 0\).
   - \(-3 < x < 0\): Positive from \(x^2\) and \((x + 3)\), negative from others, so \(f(x) < 0\).
   - \(0 < x < 3\): All factors positive, \(f(x) > 0\).
   - \(3 < x < 5\): Positive except \((x - 5)\), so \(f(x) < 0\).
   - \(x > 5\): All factors positive, \(f(x) > 0\).

5. Solution:  
\(f(x) < 0\) on \((-3, 0) \cup (3, 5)\).

\[\boxed{(-3, 0) \cup (3, 5)}\]

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### **3. Solve \(4x^2 = 3x\) by factoring.**

1. Rearrange the equation:
   \[4x^2 - 3x = 0\]

2. Factor out \(x\):
   \[x(4x - 3) = 0\]

3. Set each factor equal to zero:
   - \(x = 0\)
   - \(4x - 3 = 0 \quad \Rightarrow \quad 4x = 3 \quad \Rightarrow \quad x = \frac{3}{4}\)

4. Solution:
   \[\boxed{x = 0 \text{ or } x = \frac{3}{4}}\]

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Would you like any part of the solutions explained in more detail?  
Here are five follow-up questions to explore:  
1. How can I plot complex numbers on the complex plane?  
2. What’s the general solution pattern for inequalities involving polynomials?  
3. How do cycles of \(i\) apply to more complicated exponents?  
4. What’s another way to solve quadratic equations like \(4x^2 - 3x = 0\)?  
5. How do we know when factoring will work versus other methods like the quadratic formula?

**Tip:** Always analyze roots and their multiplicities when finding where a polynomial is positive or negative. This helps you determine if a root "bounces" or crosses the x-axis.

1. Evaluate the following: (a) (2 - 4i) - (-4 + 3i), (b) i^8. 2. Let f(x) = x^2(x - 3)^2(x - 5)(x + 3). Find the solution set for f(x) < 0 in interval notation. 3. Solve 4x^2 = 3x by factoring.

Explore step-by-step solutions for problems involving complex number operations, solving polynomial inequalities, and factoring quadratic equations. Suitable for high school students, this guide covers key mathematical concepts including powers of i, sign analysis for polynomials, and solving quadratic equations.

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