How do you foil #(6x-y)(3x-2y)#?

First terms: #6x# and #3x#
#color(white)("XXXX")##color(white)("XXXX")#Product of First terms: #18x^2#

Outside terms: #6x# and #-2y#
#color(white)("XXXX")##color(white)("XXXX")#Product of Outside terms : # -12xy#

Inside terms: #-y# and #3x#
#color(white)("XXXX")##color(white)("XXXX")#Product of Inside terms: #-3xy#

Last terms: #-y# and #-2y#
#color(white)("XXXX")##color(white)("XXXX")#Product of Last terms: #2y^2#

Sum of Products:
#color(white)("XXXX")##18x^2-12xy-3xy+2y^2#

#color(white)("XXXX")##= 18x^2-15xy+2y^2#

When we multiply two polynomials, we must multiply each (and every) term in one times each (and every) term of the other.

{Reminder: things that are to be multiplied are called "factors", things to be added are called "terms".)

Many people use FOIL to remind themselves and to keep track of how to multiply these:

Multiply: #(6x - y)(3x – 2y)" "# (in this example we'll use FOIL)

#{:(color(red)("F")"irst",(color(red)(6x) -y)(color(red)(3x) – 2y), (6x)(3x), =, 18x^2),(color(red)("O")"utside",(color(red)(6x) -y)(3x color(red)(-9)), (6x)(-2y), =, -12xy),(color(red)("I")"nside",(6xcolor(red) (-y))(color(red)(3x) – 2y), (-y)(3x), =, -3xy),(color(red)("L")"ast",(6xcolor(red) (-y))(3x color(red)(-2y)), (-y)(-2y), =, 2y^2) :}#

Written on one line, we have:

#(6x - y)(3x – 2y) = (6x)(3x)+(6x)(-2y)+(-y)(3x)+(-y)(-2y)#

#(6x - y)(3x – 2y) = 18x^2-12xy-3xy+2y^2" "# now combine similar terms

#(6x - y)(3x – 2y) = 18x^2-15xy+2y^2#