Question

determining characteristics of a rectangle
which statement is true of a rectangle that has an area of 4x² + 39x - 10 square units and a width of (x + 10) units?
the rectangle is a square.
the rectangle has a length of (2x - 5) units.
the perimeter of the rectangle is (10x + 18) units.
the area of the rectangle can be represented by (4x² + 20x - 2x - 10) square units.

Explanation:

Step1: Find the length of the rectangle

We know that the area of a rectangle $A = l\times w$, where $A = 4x^{2}+39x - 10$, $w=x + 10$. So, $l=\frac{A}{w}=\frac{4x^{2}+39x - 10}{x + 10}$.
We factor the numerator: $4x^{2}+39x - 10=4x^{2}+40x - x - 10=4x(x + 10)-(x + 10)=(4x - 1)(x + 10)$.
Then $l=\frac{(4x - 1)(x + 10)}{x + 10}=4x-1$.

Step2: Check if it is a square

Since length $l = 4x-1$ and width $w=x + 10$, $l
eq w$ for non - zero $x$, so it is not a square.

Step3: Calculate the perimeter

The perimeter $P=2(l + w)=2((4x-1)+(x + 10))=2(5x + 9)=10x+18$.

Step4: Check the area representation

$4x^{2}+20x-2x - 10=4x^{2}+18x - 10
eq4x^{2}+39x - 10$.

Answer:

The perimeter of the rectangle is $(10x + 18)$ units.