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

Step-by-step solution

1. 1

   Note what the monomial does to each term. Multiplying by $5y^{2}$ multiplies every coefficient by $5$, leaves the power of $x$ untouched, and raises the power of $y$ by exactly $2$. Keeping that rule in mind turns seven multiplications into seven small edits.

2. 2

   Handle the two terms containing x.

   $\left(x^{2}y\right)\left(5y^{2}\right)=5x^{2}y^{3},\qquad (-2x)\left(5y^{2}\right)=-10xy^{2}.$

3. 3

   Handle the constant term. A constant has $y^{0}$, so it becomes a $y^{2}$ term:

   $(-5)\left(5y^{2}\right)=-25y^{2}.$

   This is the term most often dropped — a constant is still a term.

4. 4

   Handle the four pure powers of y.

   $\left(5y^{2}\right)\left(5y^{2}\right)=25y^{4},\qquad \left(-2y^{3}\right)\left(5y^{2}\right)=-10y^{5},$
   $(6y)\left(5y^{2}\right)=30y^{3},\qquad \left(-5y^{4}\right)\left(5y^{2}\right)=-25y^{6}.$

5. 5

   Collect the seven terms in a sensible order.

   $5x^{2}y^{3}-10xy^{2}-25y^{6}-10y^{5}+25y^{4}+30y^{3}-25y^{2}.$

   The highest degree is $6$, coming from the $-5y^{4}$ term, so multiplying by $5y^2$ raised the polynomial's degree by $2$ as expected.

6. 6

   Check with a substitution. At $x=y=1$ the bracket is $1+5-5-2-2+6-5=-2$, and $5y^2=5$, so the product should be $-10$. Adding the seven answer terms at $x=y=1$: $5-10-25-10+25+30-25=-10$ ✓.