4. a triangle with base (2x+2) and height (x-2) is cut out of a rectangular sheet of paper that is (3x+4) by (2x+3) what is the area of the paper that remains?

So what we do is
area that remains=total area-triangle area that was cut out

we need to find 2 things
total area
triangle area

total area=rectange=base times height
area=(3x+4) times (2x+3)
FOIL or distribute
6x^2+8x+9x+12=6x^2+17x+12

triangle area=1/2 times base times height
triangle area=1/2 times (2x+2) times (x-2)=
(x+2) times (x-2)=x^2+2x-2x-4=x^2-4

so
total area=6x^2+17x+12
triangle area=x^2-4

subtract
area that remains=total area-triangle area that was cut out
area that remains=6x^2+17x+12-(x^2-4)=
6x^2+17x+12-x^2+4=
6x^2-x^2+17x+12+4=
5x^2+17x+16

area that remains is 5x^2+17x+16