The following questions are accompanied by two statements (A) and (B). You have to determine which statement(s) is/are sufficient /necessary to answer the questions
Find the value of `8y^(2) (y + 3x) + 8x^(2) (x + 3y) `
A gross Product of xand y is equal to 6
B. Sum-of inverse of x and y is equals to `(5)/(6)`

Options

A. if the Statement 'A' alone is sufficient to answer the question but the Statement 'B' alone is not sufficient
B. if the Statement 'B' alone is sufficient to answer the question but the Statement 'A' alone is not sufficient
C. if both Statement 'A' and 'B' together are needed to answer the question
D. if either the Statement 'A' alone or Statement 'B' alone is sufficient to answer the question

Answer

To solve the problem, we need to evaluate the expression \( 8y^2(y + 3x) + 8x^2(x + 3y) \) using the information provided in statements A and B.

### Step 1: Analyze the Statements
- **Statement A**: The gross product of \( x \) and \( y \) is equal to 6, which can be written as:
\[ xy = 6 \]

- **Statement B**: The sum of the inverses of \( x \) and \( y \) is equal to \( \frac{5}{6} \), which can be expressed as:
\[ \frac{1}{x} + \frac{1}{y} = \frac{5}{6} \]
This can be rewritten using a common denominator:
\[ \frac{y + x}{xy} = \frac{5}{6} \]

### Step 2: Combine the Statements
Using the information from both statements, we can substitute \( xy \) from Statement A into Statement B:
\[ \frac{x + y}{6} = \frac{5}{6} \]
Cross-multiplying gives:
\[ x + y = 5 \]

### Step 3: Solve for \( x \) and \( y \)
Now we have two equations:
1. \( xy = 6 \)
2. \( x + y = 5 \)

We can express \( y \) in terms of \( x \):
\[ y = 5 - x \]

Substituting this into the product equation:
\[ x(5 - x) = 6 \]
Expanding this gives:
\[ 5x - x^2 = 6 \]
Rearranging leads to:
\[ x^2 - 5x + 6 = 0 \]

### Step 4: Factor the Quadratic Equation
Factoring the quadratic:
\[ (x - 2)(x - 3) = 0 \]
Thus, the solutions are:
\[ x = 2 \quad \text{or} \quad x = 3 \]

### Step 5: Find Corresponding \( y \) Values
Using \( y = 5 - x \):
- If \( x = 2 \), then \( y = 3 \).
- If \( x = 3 \), then \( y = 2 \).

### Step 6: Substitute Back into the Expression
Now we can substitute \( x \) and \( y \) back into the original expression:
\[ 8y^2(y + 3x) + 8x^2(x + 3y) \]

Calculating for \( (x, y) = (2, 3) \):
\[ = 8(3^2)(3 + 3 \cdot 2) + 8(2^2)(2 + 3 \cdot 3) \]
\[ = 8(9)(3 + 6) + 8(4)(2 + 9) \]
\[ = 8(9)(9) + 8(4)(11) \]
\[ = 648 + 352 = 1000 \]

Calculating for \( (x, y) = (3, 2) \):
\[ = 8(2^2)(2 + 3 \cdot 3) + 8(3^2)(3 + 3 \cdot 2) \]
\[ = 8(4)(2 + 9) + 8(9)(3 + 6) \]
\[ = 8(4)(11) + 8(9)(9) \]
\[ = 352 + 648 = 1000 \]

### Conclusion
Thus, the value of the expression \( 8y^2(y + 3x) + 8x^2(x + 3y) \) is \( 1000 \).

### Final Answer
Both statements A and B are necessary to determine the value of the expression.