To find the area of the rectangle, we need to multiply the height by the width. The height is given as $3b^2$, and the width is $4b^4 + 2b - 6$.

Step 1: Write the expression for the area.

The area $A$ is given by:

$A = \text{Height} \times \text{Width} = 3b^2 \times (4b^4 + 2b - 6)$

Step 2: Distribute the height over each term of the width.

Now, distribute $3b^2$ to each term inside the parentheses:

$A = 3b^2 \times 4b^4 + 3b^2 \times 2b + 3b^2 \times (-6)$

Step 3: Simplify each term.

1. $3b^2 \times 4b^4 = 12b^6$
2. $3b^2 \times 2b = 6b^3$
3. $3b^2 \times (-6) = -18b^2$

Step 4: Combine the terms.

Thus, the area is:

$A = 12b^6 + 6b^3 - 18b^2$

This is the area of the rectangle in polynomial form.

Final Answer:

$A = 12b^6 + 6b^3 - 18b^2$

Would you like further details on any part of the process or have any other questions?

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Here are some related questions:

1. How would you find the area of a rectangle if the height and width were polynomials with different degrees?
2. How can you factor the polynomial expression $12b^6 + 6b^3 - 18b^2$?
3. What is the degree of the polynomial $12b^6 + 6b^3 - 18b^2$?
4. How do you multiply polynomials with different numbers of terms?
5. Can you calculate the perimeter of this rectangle if given the same height and width?

Tip: When distributing terms, always keep track of the exponents and apply the power rule of exponents: $a^m \times a^n = a^{m+n}$.