https://r2.erweima.ai/mathbot/image/1726719390655wvzdxqr8.jpg

The expression provided in the image is:

\[
(4a^4 - 2a^3b + 5a^2b^2 - 3ab^3 + b^4)(a^2 - ab - b^2)
\]

This is a polynomial multiplication problem, where we are tasked with multiplying two polynomials:

- \( 4a^4 - 2a^3b + 5a^2b^2 - 3ab^3 + b^4 \)
- \( a^2 - ab - b^2 \)

To solve this, we need to distribute each term of the first polynomial over each term of the second polynomial and combine like terms.

Let's proceed step by step.

### Step 1: Expand the expression

We multiply each term of the first polynomial by each term of the second polynomial.

\[
\begin{aligned}
(4a^4)(a^2) & = 4a^6, \\
(4a^4)(-ab) & = -4a^5b, \\
(4a^4)(-b^2) & = -4a^4b^2, \\
\\
(-2a^3b)(a^2) & = -2a^5b, \\
(-2a^3b)(-ab) & = 2a^4b^2, \\
(-2a^3b)(-b^2) & = 2a^3b^3, \\
\\
(5a^2b^2)(a^2) & = 5a^4b^2, \\
(5a^2b^2)(-ab) & = -5a^3b^3, \\
(5a^2b^2)(-b^2) & = -5a^2b^4, \\
\\
(-3ab^3)(a^2) & = -3a^3b^3, \\
(-3ab^3)(-ab) & = 3a^2b^4, \\
(-3ab^3)(-b^2) & = 3ab^5, \\
\\
(b^4)(a^2) & = a^2b^4, \\
(b^4)(-ab) & = -ab^5, \\
(b^4)(-b^2) & = -b^6.
\end{aligned}
\]

### Step 2: Combine like terms

Now, let's group terms with the same powers of \( a \) and \( b \).

\[
\begin{aligned}
& 4a^6 \\
& - (4a^5b + 2a^5b) = -6a^5b \\
& (-4a^4b^2 + 2a^4b^2 + 5a^4b^2) = 3a^4b^2 \\
& (2a^3b^3 - 5a^3b^3 - 3a^3b^3) = -6a^3b^3 \\
& (-5a^2b^4 + 3a^2b^4 + a^2b^4) = -a^2b^4 \\
& (3ab^5 - ab^5) = 2ab^5 \\
& -b^6.
\end{aligned}
\]

### Final Answer:

\[
4a^6 - 6a^5b + 3a^4b^2 - 6a^3b^3 - a^2b^4 + 2ab^5 - b^6
\]

This is the expanded form of the given polynomial expression.

Would you like further details or have any questions? 

Here are 5 related questions for deeper exploration:

1. How would you factor the expanded polynomial?
2. Can this expression be simplified further, considering any common factors between terms?
3. How does the degree of each term change after expansion?
4. What happens if you set \( a = b \) in the original expression?
5. How would you visualize the terms of this polynomial geometrically?

**Tip**: Always keep an eye on the coefficients and powers while multiplying polynomials to avoid missing any terms or combining incorrectly.

(4a^4 - 2a^3b + 5a^2b^2 - 3ab^3 + b^4)(a^2 - ab - b^2)

Explore the step-by-step expansion of the polynomial expression (4a^4 - 2a^3b + 5a^2b^2 - 3ab^3 + b^4)(a^2 - ab - b^2). This problem demonstrates polynomial multiplication using the distributive property, suitable for students in Grades 10-12.

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