If the original frame area is x, then length*width=12*10=120=area=x. By tripling the frame's area, we get 120*3=360. However, we need to add a number x to both the length and width to equal the new area. Since length*width=area, and 10+x is the width and 12+x is the length, we get (10+x)*(12+x)=360. Expanding using the Distributive Property, we get 120+10x+12x+x^2=x^2+22x+120=360. Subtracting 360 from both sides, we get x^2+22x-240=0.Finding 2 numbers that add up to 22 and multiply to -240, we do a bit of guess and check ( we COULD use the quadratic formula too) to find 30 and -8 fit the bill. Therefore, x^2+30x-8x-240=0=
x(x+30)-8(x+30)=(x-8)(x+30). To make the expression equal to 0, since 0 multiplied by anything is 0, we have to figure out when x-8 and x+30 equal 0, which is when x=8 and -30 respectively (To do this, you could add 8 from both sides to turn x-8=0 to x=8, and something similar to -30). Plugging each one in, we get (10+(8))*(12+(8))=18*20=360. To make sure there aren't 2 solutions, we get that the new width is 10-30=-20, which clearly can't be negative , making x equal to 8.
