We need to expand this problem and then combine like-terms to simplify it
I'd like to break this down:
#color(red)((3u+v)) * color(green)((3u+v)) * color(blue)((3u+v)) * color(purple)((3u+v))#
Just to make this easier, I'm going to do this one set at a time:
#color(red)((3u+v)) * color(green)((3u+v))#
#3u * 3u + 3u*v + v*3u + v * v#
#9u^2 + 6uv + v^2#
#color(white)(0)#
#color(blue)((3u+v)) * color(purple)((3u+v))#
#3u * 3u + 3u*v + v*3u + v * v#
#9u^2 + 6uv + v^2#
#color(white)(0)#
Now we have
#(9u^2 + 6uv + v^2) xx (9u^2 + 6uv + v^2)#
I like to put parentheses around the factors so I don't mess up the addition with the multiplication
#(9u^2 xx 9u^2) + (9u^2 xx 6uv )+ (9u^2*v^2 )+ (6uv xx 9u^2) + (6uv xx 6uv) + (6uv xx v^2) + (v^2 xx 9u^2) + (v^2 xx 6uv) + (v^2 xx v^2)#
#81u^4 + 54u^3v + 9u^2v^2 + 54u^3v + 36u^2v^2 + 6v^3u + 9u^2v^2 + 6v^3u + v^4#
rearrange this to put like-terms next to each other
#81u^4 +
(54u^3v + 54u^3v) +
(9u^2v^2 + 36u^2v^2 + 9u^2v^2) +
(6v^3u + 6v^3u)
v^4#
#81u^4 + 108u^3v + 54u^2v^2 + 12v^3u + v^4#
