Answer
[tex](12+x)(15+x)=160+15*12[/tex]
Which simplifies to:
[tex]x^2+27x-160=0[/tex]
Explanation
Let [tex]x[/tex] the number of feet they are going to expand the width and length of the patio. So the new width of the patio will be [tex]12+x[/tex] and the new length [tex]15+x[/tex].
Remember that the area of a rectangle is width times length, so the length of our expanded patio will be [tex](12+x)(15+x)[/tex]. But, we also know form our problem that they want to increase the total area of the patio by 160 ft2. The actual area of the patio is [tex](15ft)(12ft)[/tex], so the expanded area of the patio will be [tex]160ft^2+(15ft*12ft)[/tex].
Now we can equate the expanded area of the patio and simplify:
[tex](12+x)(15+x)=160+15*12[/tex]
[tex]180+12x+15x+x^2=160+180[/tex]
[tex]180+27x+x^2=340[/tex]
[tex]x^2+27x-160=0[/tex]
