First, let's mark each section of our equation in color coding, so we can more easily recognize where each term comes from.
#color(red)((y+2)^3) color(blue)(-(y+3)(y-3)) color(green)( - 2(y-2)^2) color(orange)(-(2y-1-y^2)^2) color(pink)(-y^2(-y^2+5y-3))#
Next, I'm going to simplify this step-by-step. First, we'll simplify #color(red)("red")#. For this, we simply distribute the#color(white)(.)^3#to the two terms inside the parenthesis, giving us:
#color(red)((y^3+8))#
Now we simplify #color(blue)("blue")# by multiplying the #y# of the first term by the #y#, then #-3# of the second term. Then, repeat with the #3# of the first term.
So first, the #y#
#color(blue)(y * y = y^2)#
#color(blue)(y * (-3) = (-3y))#
Now for the #3#
#color(blue)(3 * y = 3y)#
#color(blue)(3 * (-3) = (-9)#
Put these in order in a new term, remembering to keep our negative from subtraction, which says:
#color(blue)(-(y^2 cancel(- 3y + 3y) - 9)) rArr color(blue)(-(y^2-9))#
Now we simplify the #color(green)("green")#. This is very similar to how we handled #color(blue)("blue")#.
#color(green)(-2(y-2)^2) rArr color(green)((-2y+4)^2) rArr color(green)((-2y+4)(-2y+4))#
#"---"#
#color(green)((-2y) * (-2y)=4y^2)#
#color(green)((-2y) * 4=(-8y))#
#"---"#
#color(green)(4 * (-2y)=(-8y))#
#color(green)(4 * 4=16)#
Thus, #color(green)((4y^2-16y+16))#
Now #color(orange)("orange")#, similar again to #color(blue)("blue")# and #color(green)("green")#.
#color(orange)((-2y-1-y^2)^2) rArr color(orange)((-2y-1-y^2)(-2y-1-y^2))#
#"---"#
#color(orange)((-2y) * (-2y) = 4y^2)#
#color(orange)((-2y) * (-1) = 2y#
#color(orange)((-2y) * (-y^2) = 2y^3#
#"---"#
#color(orange)((-1) * (-2y) = 2y#
#color(orange)((-1) * (-1) = 1#
#color(orange)((-1) * (-y^2) = y^2#
#"---"#
#color(orange)((-y^2) * (-2y) = 2y^3#
#color(orange)((-y^2) * (-1) = y^2#
#color(orange)((-y^2) * (-y^2) = y^4#
#"---"#
#color(orange)((4y^2 + 2y + 2y^3+2y+1+y^2+2y^3+y^2+y^4)#
Combine like terms to get:
#color(orange)((y^4+4y^3+6y^2+4y+1))#
And finally, #color(pink)("pink")#.
#color(pink)(-y^2(-y^2+5y-3)) rArr color(pink)((y^4-5y^3+3y^2))#
Now, put all of your answers from this together, and drop the parenthesis.
#color(red)(y^3+8) color(blue)(-y^2-9) color(green)(+ 4y^2 - 16y + 16) color(orange)(+ y^4+4y^3+6y^2+4y+1) color(pink)(+y^4-5y^3+3y^2)#
Finally, combine your like terms.
#2y^4 + 10y^3 + 12y^2 - 12y + 16#
