Part A: From the diagram of the room, it is clear that the width is w=x+4.
Let l be the length of the room so it follows:
[tex]\begin{gathered} l=2x+6+x-4 \\ l=(2x+x)+(6-4) \\ l=3x+2 \end{gathered}[/tex]
Hence the expression for the total area of the room if the area is A is given by:
[tex]\begin{gathered} A=lw \\ A=(3x+2)(x+4) \\ A=3x^2+2x+12x+8 \\ A=3x^2+14x+8 \end{gathered}[/tex]
Part B: It is given that the length is 12 units so l=12, therefore it follows:
[tex]\begin{gathered} 12=3x+2 \\ 12-2=3x-2 \\ 10=3x \\ x=\frac{10}{3} \end{gathered}[/tex]
So the area from part A at x=10/3 will be:
[tex]\begin{gathered} A=3x^2+14x+8 \\ A=3(\frac{10}{3})^2+14(\frac{10}{3})+8 \\ A=3\times\frac{100}{9}+\frac{140}{3}+8 \\ A=\frac{300}{9}+\frac{140\times3}{3\times3}+\frac{8\times9}{1\times9} \\ A=\frac{300+420+72}{9} \\ A=\frac{792}{9} \\ A=88\text{ sq ft} \end{gathered}[/tex]
Hence the total area of both rooms is 88 square feet.
