Question

the total area of the figure to the right is 465 $\text{cm}^2$. use this fact to write an equation involving $x$. then solve the equation to find the value of $x$.
the equation is $square$

Explanation:

Step1: Define total area equation

The figure can be seen as a larger rectangle with dimensions $(20 + x)$ and $(8 + x)$, minus the inner empty rectangle. The area of the larger rectangle is $(20+x)(8+x)$, the area of the inner rectangle is $20 \times 8$, and their difference equals the total area 465.
$$(20+x)(8+x) - 20\times8 = 465$$

Step2: Expand and simplify equation

Expand the product and compute constants:
$$160 + 20x + 8x + x^2 - 160 = 465$$
$$x^2 + 28x = 465$$
$$x^2 + 28x - 465 = 0$$

Step3: Solve quadratic equation

Factor the quadratic or use quadratic formula. Factoring gives:
$$(x+37)(x-5) = 0$$
Since length cannot be negative, we take the positive root:
$$x = 5$$

Answer:

$x = 5$