Answer
a) $4x^2+44x$
b) $3$ feet
Work Step by Step
a) The dimensions of the large rectangle are $2x+12$ and $2x+10$. We are given the dimensions of the small rectangle: $12$ and $10$. We will write the area of the border as the difference between the area of the big rectangle and the area of the small rectangle:
$$\begin{align*}
\text{Area}&=(2x+12)(2x+10)-12\cdot 10\\
&=4x^2+20x+24x+120-120\\
&=4x^2+44x.
\end{align*}$$
b) We solve the equation $\text{Area}=168$ for $x$:
$$\begin{align*}
4x^2+44x&=168\quad&&\text{Write the given equation.}\\
4x^2+44x-168&=0\quad&&\text{Subtract }168\text{ from each side.}\\
4(x^2+11x-42)&=0\quad&&\text{Factor out }4.\\
4(x-3)(x+14)&=0\quad&&\text{Factor the trinomial.}\\
x-3=0&\text{ or }x+14=0\quad&&\text{Set each factor equal to 0.}\\
x=3&\text{ or }x=-14\quad&&\text{Solve the resulting equations.}
\end{align*}$$
As $x$ is a dimension, it must be positive, so the only solution is $x=3$. So the width of the strip is $3$ feet.
