Two different container contains `(2x^(3) + 2x^(2) + 3x + 3)` L and `(4x^(3) – 2x^(2) + 6x - 3)` L water, What is biggest measure that can measure both quantities exactly?

Options

A. `(x^(2) 2x)L`
B. `(2x^(2) + 3 ) L`
C. `(2x-1) L`
D. `(x+1) L`

Answer

To find the biggest measure that can measure both quantities of water in the two containers exactly, we need to determine the highest common factor (HCF) of the two polynomial expressions representing the volumes of water in the containers.

### Step-by-Step Solution:

1. **Identify the Polynomials:**
- The first container has a volume of \( P_1 = 2x^3 + 2x^2 + 3x + 3 \) liters.
- The second container has a volume of \( P_2 = 4x^3 - 2x^2 + 6x - 3 \) liters.

2. **Factor the First Polynomial \( P_1 \):**
- We can factor \( P_1 \) by grouping:
\[ P_1 = 2x^3 + 2x^2 + 3x + 3 \]
Grouping the terms:
\[ = (2x^3 + 2x^2) + (3x + 3) \]
Factoring out common terms:
\[ = 2x^2(x + 1) + 3(x + 1) \]
Now, factor out \( (x + 1) \):
\[ = (2x^2 + 3)(x + 1) \]

3. **Factor the Second Polynomial \( P_2 \):**
- Similarly, we can factor \( P_2 \):
\[ P_2 = 4x^3 - 2x^2 + 6x - 3 \]
Grouping the terms:
\[ = (4x^3 - 2x^2) + (6x - 3) \]
Factoring out common terms:
\[ = 2x^2(2x - 1) + 3(2x - 1) \]
Now, factor out \( (2x - 1) \):
\[ = (2x^2 + 3)(2x - 1) \]

4. **Find the HCF of the Two Polynomials:**
- The factored forms are:
\[ P_1 = (2x^2 + 3)(x + 1) \]
\[ P_2 = (2x^2 + 3)(2x - 1) \]
- The common factor in both polynomials is \( (2x^2 + 3) \).

5. **Conclusion:**
- Therefore, the biggest measure that can measure both quantities exactly is:
\[ \text{Biggest Measure} = 2x^2 + 3 \text{ liters} \]