###### The problem

A rectangular piece of paper has length (x +5) cm and width ( x +2 ) cm.

A square with sides of length, x cm is removed.

a) Write an expression for the area of the rectangle before the square is cut out. Expand the brackets.

b) Write an expression for the shaded area

c) Find x if the shaded area is 31$c{m}^{2}$.

###### The solution

a) Expression for the area of the rectangle before the square is cut.

( x + 5) ( x + 2) $a)(x+5)(x+2)={x}^{2}+7x+10$

$b)Expressionofshadedarea\phantom{\rule{0ex}{0ex}}({x}^{2}+7x+10)-{x}^{2}={x}^{2}+7x+10-{x}^{2}=7x+10.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}c)Findxiftheshadedareais31c{m}^{2}.\phantom{\rule{0ex}{0ex}}7x+10=31\Rightarrow 7x=31-10=21\phantom{\rule{0ex}{0ex}}\Rightarrow 7x=21\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{21}{7}=3\Rightarrow x=3.$