Divide the polynomial by the monomial: (20x^2y^2 + 16x^2y - 12xy^2) divided by 4xy. Check the answer by showing that the product of the divisor and the quotient is the dividend.

Step-by-step solution

1. 1

   Split the single fraction into one fraction per term. Division distributes over addition in the numerator (though never over the denominator):

   $\frac{20x^{2}y^{2}}{4xy}+\frac{16x^{2}y}{4xy}-\frac{12xy^{2}}{4xy}.$

2. 2

   Divide the first term. Handle the coefficient and each variable separately, subtracting exponents:

   $\frac{20}{4}=5,\qquad \frac{x^{2}}{x}=x,\qquad \frac{y^{2}}{y}=y\ \Longrightarrow\ 5xy.$

3. 3

   Divide the second term. Here the $y$ powers cancel completely, leaving $y^{0}=1$:

   $\frac{16}{4}=4,\qquad \frac{x^{2}}{x}=x,\qquad \frac{y}{y}=1\ \Longrightarrow\ 4x.$

4. 4

   Divide the third term, keeping its sign. The $x$ powers cancel this time:

   $\frac{-12}{4}=-3,\qquad \frac{x}{x}=1,\qquad \frac{y^{2}}{y}=y\ \Longrightarrow\ -3y.$

5. 5

   Assemble the quotient.

   $5xy+4x-3y.$

6. 6

   Carry out the requested check. Multiply the divisor by the quotient:

   $4xy\left(5xy+4x-3y\right)=20x^{2}y^{2}+16x^{2}y-12xy^{2},$

   which is exactly the dividend, so the division is correct (for $x\neq 0$, $y\neq 0$).