To simplify the expression $(a^2 - 2b^2)(a^2 + b^2) + 4(3a^2 - b^2)(b^2 - a)$, let’s break it down systematically:

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Step 1: Expand the first term:

$(a^2 - 2b^2)(a^2 + b^2)$ Using the distributive property: $= a^4 + a^2b^2 - 2b^2a^2 - 2b^4$ Simplify the middle terms: $= a^4 - a^2b^2 - 2b^4$

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Step 2: Expand the second term:

$4(3a^2 - b^2)(b^2 - a)$ Using the distributive property: $= 4[(3a^2)(b^2) - (3a^2)(a) - (b^2)(b^2) + (b^2)(a)]$ $= 4[3a^2b^2 - 3a^3 - b^4 + ab^2]$ Distribute the 4: $= 12a^2b^2 - 12a^3 - 4b^4 + 4ab^2$

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Step 3: Combine both expanded terms:

Now, add the results of the two expansions: $(a^4 - a^2b^2 - 2b^4) + (12a^2b^2 - 12a^3 - 4b^4 + 4ab^2)$ Group similar terms:

• $a^4$: $a^4$
• $a^3$: $-12a^3$
• $a^2b^2$: $-a^2b^2 + 12a^2b^2 = 11a^2b^2$
• $b^4$: $-2b^4 - 4b^4 = -6b^4$
• $ab^2$: $4ab^2$

So the final simplified expression is: $a^4 - 12a^3 + 11a^2b^2 - 6b^4 + 4ab^2$

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Final Answer:

$a^4 - 12a^3 + 11a^2b^2 - 6b^4 + 4ab^2$

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Do you want a detailed breakdown of any specific step?

5 Related Questions:

1. How can we factorize the resulting expression further?
2. What if $a = b$, how does the expression simplify?
3. What role does distributive property play in polynomial expansions?
4. Can this expression be represented geometrically?
5. How would this simplify if $b^2 = ka^2$, where $k$ is a constant?

Tip:

When simplifying, always group terms by powers and coefficients systematically to avoid missing simplifications!