The product of additive inverse of `(x+6)/(x+2)` and `(5x+2)/(5x-3)` is:

Options

A. `(5x^(2) + 32x+12)/(5x^(2) - 7x + 6)`
B. `(5x^(2) + 32 x + 12)/(5x^(2) + 7x -6)`
C. `(5x^(2) + 32 x + 12)/(5x^(2) + 7x + 6)`
D. None of these

Answer

To find the product of the additive inverse of \(\frac{x+6}{x+2}\) and \(\frac{5x+2}{5x-3}\), we can follow these steps:

### Step 1: Find the Additive Inverse
The additive inverse of a fraction \(\frac{a}{b}\) is given by \(-\frac{a}{b}\). Therefore, the additive inverse of \(\frac{x+6}{x+2}\) is:
\[ -\frac{x+6}{x+2} \]

### Step 2: Write the Second Expression
The second expression is:
\[ \frac{5x+2}{5x-3} \]

### Step 3: Multiply the Two Expressions
Now we need to multiply the additive inverse of the first expression by the second expression:
\[ -\frac{x+6}{x+2} \cdot \frac{5x+2}{5x-3} \]

### Step 4: Combine the Fractions
When multiplying fractions, we multiply the numerators together and the denominators together:
\[ = -\frac{(x+6)(5x+2)}{(x+2)(5x-3)} \]

### Step 5: Expand the Numerator
Now we will expand the numerator:
\[ (x+6)(5x+2) = x \cdot 5x + x \cdot 2 + 6 \cdot 5x + 6 \cdot 2 \]
\[ = 5x^2 + 2x + 30x + 12 \]
\[ = 5x^2 + 32x + 12 \]

### Step 6: Expand the Denominator
Now we will expand the denominator:
\[ (x+2)(5x-3) = x \cdot 5x + x \cdot (-3) + 2 \cdot 5x + 2 \cdot (-3) \]
\[ = 5x^2 - 3x + 10x - 6 \]
\[ = 5x^2 + 7x - 6 \]

### Step 7: Combine the Results
Now we can combine the results from the numerator and denominator:
\[ = -\frac{5x^2 + 32x + 12}{5x^2 + 7x - 6} \]

### Final Result
Thus, the product of the additive inverse of \(\frac{x+6}{x+2}\) and \(\frac{5x+2}{5x-3}\) is:
\[ -\frac{5x^2 + 32x + 12}{5x^2 + 7x - 6} \]