Expand (8 - 6xy - x^2y^2 - 6x^3 - 5y^3 - 7y + 4x) times 8xy.

Step-by-step solution

1. 1

   Plan the bookkeeping before multiplying. There are seven terms inside the bracket, so there will be seven terms in the answer — no combining is possible afterwards, because each product ends up with a different pair of exponents. Work through the bracket in order so nothing is skipped.

2. 2

   Multiply the coefficients and add the exponents separately. For each term, multiply the numerical coefficients, add the powers of $x$, and add the powers of $y$. Remember every bare $x$ or $y$ carries an invisible exponent $1$:

   $8\cdot 8xy=64xy,\qquad (-6xy)(8xy)=-48x^{2}y^{2},\qquad \left(-x^{2}y^{2}\right)(8xy)=-8x^{3}y^{3}.$

3. 3

   Continue with the pure-power terms. These pick up a factor of the other variable from the monomial:

   $\left(-6x^{3}\right)(8xy)=-48x^{4}y,\qquad \left(-5y^{3}\right)(8xy)=-40xy^{4}.$

4. 4

   Finish the two remaining terms.

   $(-7y)(8xy)=-56xy^{2},\qquad (4x)(8xy)=32x^{2}y.$

5. 5

   Assemble the answer in descending order of the x-power.

   $-48x^{4}y-8x^{3}y^{3}-48x^{2}y^{2}+32x^{2}y-40xy^{4}-56xy^{2}+64xy.$

6. 6

   Check with a substitution. At $x=y=1$ the bracket is $8-6-1-6-5-7+4=-13$ and the monomial is $8$, so the product is $-104$. Adding the seven answer terms at $x=y=1$: $-48-8-48+32-40-56+64=-104$ ✓.