Divide `36x^(2)y^(5)+42xy^(3)-24x^(3)y^(2)-12y^(5)` by `-6y^(2)`.

Options

A. `6x^(2)y^(3)-7xy+4x^(3)+2y^(3)`
B. `-6x^(2)y^(3)-7xy^(2)+4x^(2)+2y^(3)`
C. `-6x^(2)y^(3)-7xy+4x^(3)+2y^(3)`
D. none of these

Answer

To solve the problem of dividing \( 36x^2y^5 + 42xy^3 - 24x^3y^2 - 12y^5 \) by \( -6y^2 \), we will follow these steps:

### Step 1: Set up the division
We need to divide each term of the polynomial \( 36x^2y^5 + 42xy^3 - 24x^3y^2 - 12y^5 \) by \( -6y^2 \).

### Step 2: Divide the first term
Divide \( 36x^2y^5 \) by \( -6y^2 \):
\[ \frac{36x^2y^5}{-6y^2} = -6x^2y^{5-2} = -6x^2y^3 \]

### Step 3: Divide the second term
Divide \( 42xy^3 \) by \( -6y^2 \):
\[ \frac{42xy^3}{-6y^2} = -7xy^{3-2} = -7xy \]

### Step 4: Divide the third term
Divide \( -24x^3y^2 \) by \( -6y^2 \):
\[ \frac{-24x^3y^2}{-6y^2} = 4x^3 \]

### Step 5: Divide the fourth term
Divide \( -12y^5 \) by \( -6y^2 \):
\[ \frac{-12y^5}{-6y^2} = 2y^{5-2} = 2y^3 \]

### Step 6: Combine the results
Now, we combine all the results from the divisions:
\[ -6x^2y^3 - 7xy + 4x^3 + 2y^3 \]

### Final Answer
Thus, the result of dividing \( 36x^2y^5 + 42xy^3 - 24x^3y^2 - 12y^5 \) by \( -6y^2 \) is:
\[ -6x^2y^3 - 7xy + 4x^3 + 2y^3 \]