If a, equals, 5, plus, 4, ia=5+4i and b, equals, minus, 3, xb=−3x, then find the value of the a, squared, ba 
2
 b in fully simplified form

We are tasked with finding \( a^2b \) where:

- \( a = 5 + 4i \) (a complex number)
- \( b = -3x \) (assuming this represents a real number multiplied by a variable \( x \))

We will calculate \( a^2 \) first, then multiply the result by \( b = -3x \).

### Step 1: Calculate \( a^2 \)
Given \( a = 5 + 4i \), we find \( a^2 \) by squaring the complex number:

\[
a^2 = (5 + 4i)^2
\]
Using the formula \( (x + yi)^2 = x^2 + 2xyi - y^2 \), where \( x = 5 \) and \( y = 4 \):

\[
a^2 = 5^2 + 2(5)(4)i + (4i)^2
\]
\[
a^2 = 25 + 40i + 16i^2
\]

Since \( i^2 = -1 \), this simplifies to:

\[
a^2 = 25 + 40i - 16
\]
\[
a^2 = 9 + 40i
\]

### Step 2: Multiply \( a^2 \) by \( b \)
Now we multiply \( a^2 = 9 + 40i \) by \( b = -3x \):

\[
a^2b = (9 + 40i)(-3x)
\]
Distribute \( -3x \) over the terms in \( 9 + 40i \):

\[
a^2b = 9(-3x) + 40i(-3x)
\]
\[
a^2b = -27x - 120xi
\]

Thus, the fully simplified form of \( a^2b \) is:

\[
a^2b = -27x - 120xi
\]

### Final Answer:
\[
a^2b = -27x - 120xi
\]

Would you like further details or have any questions about this problem?

Here are 5 related questions to explore:

1. How do you square a complex number in general form?
2. What are the properties of imaginary numbers, specifically \( i^2 \)?
3. How do you simplify expressions involving both real and imaginary components?
4. How does the distribution of a real scalar over a complex number work?
5. Can you find the product of two complex numbers?

**Tip:** When multiplying or squaring complex numbers, always simplify \( i^2 \) to \( -1 \) early in your calculations to avoid mistakes.

If a = 5 + 4i and b = -3x, then find the value of a²b in fully simplified form.

Learn how to find the product a²b where a = 5 + 4i and b = -3x. This complex number problem covers algebraic manipulation, squaring complex numbers, and applying distributive properties. Perfect for high school students studying complex numbers and imaginary units.

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