Find the product (x^2 + y^2)(2x^2 + 4y^2 + 6z^2), writing the multiplication horizontally.

Step-by-step solution

1. 1

   Plan the distribution. Two terms times three terms gives six monomials. With three variables in play, "like terms" no longer means "same power of $x$" — it means the same variable pattern, e.g. $x^2y^2$ matches only another $x^2y^2$.

2. 2

   Distribute $x^2$. $x^2\cdot 2x^2=2x^4$, $x^2\cdot 4y^2=4x^2y^2$, and $x^2\cdot 6z^2=6x^2z^2$.

3. 3

   Distribute $y^2$. $y^2\cdot 2x^2=2x^2y^2$, $y^2\cdot 4y^2=4y^4$, and $y^2\cdot 6z^2=6y^2z^2$.

4. 4

   Sort the six monomials by pattern. $x^4$: one term, $2x^4$. $x^2y^2$: two terms, $4x^2y^2$ and $2x^2y^2$. $x^2z^2$: one term. $y^4$: one term. $y^2z^2$: one term.

5. 5

   Combine the single matching pair. $4x^2y^2+2x^2y^2=6x^2y^2$, so the product is $2x^4+6x^2y^2+6x^2z^2+4y^4+6y^2z^2$.

6. 6

   Check with numbers. At $x=2,y=1,z=3$: the factors are $4+1=5$ and $8+4+54=66$, product $330$; the expansion gives $32+24+216+4+54=330$. At $x=1,y=2,z=0$: $(1+4)(2+16)=90$ and $2+24+0+64+0=90$.