Match Equivalent Expressions: (4+x)(y+8+x) and Related Terms

To solve this problem, we'll expand each expression and find which polynomial they correspond with:

Start with expression 1: $(4+x)(y+8+x)$.

• Using the distributive property, expand as follows:
  $= 4(y+8+x) + x(y+8+x)$
  $= 4y + 32 + 4x + xy + 8x + x^2$
  $= x^2 + xy + 12x + 4y + 32$

This matches the polynomial $x^2 + 12x + xy + 4y + 32$, which is option b.

Next, consider expression 2: $(4+x+y)(8+x)$.

• Once again, expand using distributive property:
  $= (4+x+y)8 + (4+x+y)x$
  $= 32 + 8x + 8y + 4x + x^2 + xy$
  $= x^2 + 12x + xy + 8y + 32$

This matches the polynomial $x^2 + 12x + xy + 8y + 32$, which is option c.

Finally, consider expression 3: $(12+x)(y+x)$.

• Use distributive property to expand:
  $= 12(y+x) + x(y+x)$
  $= 12y + 12x + xy + x^2$
  $= x^2 + 12x + xy + 12y$

This matches the polynomial $x^2 + 12x + xy + 12y$, which is option a.

The correct matches are therefore: 1-c, 2-b, 3-a.