Compare the following two quantities:
**Quantity I:** $$\frac{(2a^3 + 2b^3)(a-b)}{(6a^2 - 7ab + 2b^2 + 5ab - 4a^2)(a^2 - b^2)}$$
**Quantity II:** Value of $$P - 2Q$$ where $$Q

Question

Compare the following two quantities:
**Quantity I:** $$\frac{(2a^3 + 2b^3)(a-b)}{(6a^2 - 7ab + 2b^2 + 5ab - 4a^2)(a^2 - b^2)}$$
**Quantity II:** Value of $$P - 2Q$$ where $$Q < -1 < P$$ and $$P, Q$$ are integers.
Determine the correct relation between Quantity I and Quantity II.

Accepted Answer

**Solution:** 1. **Simplify Quantity I:** Given: $$ ext{Quantity I} = \frac{(2a^3 + 2b^3)(a-b)}{(6a^2 - 7ab + 2b^2 + 5ab - 4a^2)(a^2 - b^2)}$$ First, simplify the denominator's first bracket: $$6a^2 - 7ab + 2b^2 + 5ab - 4a^2 = (6a^2 - 4a^2) + (-7ab + 5ab) + 2b^2 = 2a^2 - 2ab + 2b^2$$ So Quantity I becomes: $$\frac{(2a^3 + 2b^3)(a-b)}{(2a^2 - 2ab + 2b^2)(a^2 - b^2)}$$ 2. Factor and simplify numerator and denominator: - Factor numerator: $$2(a^3 + b^3)(a-b)$$ - Factor denominator: $$2(a^2 - ab + b^2)(a^2 - b^2)$$ Recall that: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ and $$a^2 - b^2 = (a - b)(a + b)$$ Substitute these: $$\frac{2(a + b)(a^2 - ab + b^2)(a - b)}{2(a^2 - ab + b^2)(a - b)(a + b)}$$ 3. Cancel common terms: $$2, (a + b), (a^2 - ab + b^2), (a - b)$$ Quantity I simplifies to: $$1$$ 4. **Analyze Quantity II:** Given: $$Q < -1 < P$$ where $$P, Q$$ are integers. Since $$Q < -1$$, the smallest integer less than -1 is at most -2 or less. Since $$P > -1$$, the smallest integer greater than -1 is 0 or more. Calculate $$P - 2Q$$: - Since $$Q < -1$$, $$Q \leq -2$$ - So, $$-2Q \geq -2(-2) = 4$$ (because multiplying by negative reverses inequality) - Since $$P > -1$$, $$P \geq 0$$ Therefore, $$P - 2Q \geq 0 + 4 = 4$$ So Quantity II is at least 4. 5. **Compare:** - Quantity I = 1 - Quantity II ≥ 4 Hence, $$ ext{Quantity I} < ext{Quantity II}$$ Also, since Quantity I = 1 and Quantity II ≥ 4, Quantity I ≤ Quantity II is also true. **Answer:** Options (a) and (d) are correct: Quantity I < Quantity II and Quantity I ≤ Quantity II.

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