Question

find the formula for the area of the shaded region and express it in factored form. the area of the shaded region is (factor completely.)

Explanation:

Step1: Calculate area of large square

The side - length of the large square is $9x$. The area of a square is $A = s^2$, so the area of the large square is $(9x)^2=81x^{2}$.

Step2: Calculate area of one small square

The side - length of each small square is 2. The area of a small square is $A = s^2$, so the area of one small square is $2^2 = 4$.

Step3: Calculate total area of 4 small squares

There are 4 small squares. The total area of the 4 small squares is $4\times4=16$.

Step4: Calculate area of shaded region

The area of the shaded region is the area of the large square minus the total area of the 4 small squares. So, $A=81x^{2}-16$.

Step5: Factor the expression

We use the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$. Here, $a = 9x$ and $b = 4$. So, $81x^{2}-16=(9x + 4)(9x - 4)$.

Answer:

$(9x + 4)(9x - 4)$