Question

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

Explanation:

Step1: Find the length of the rectangle

We know that the area of a rectangle \(A = l\times w\), where \(A = 4x^{2}+39x - 10\) and \(w=x + 10\).
We use polynomial long - division or factoring.
We factor \(4x^{2}+39x - 10\):

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Since \(A=l\times w=(4x - 1)(x + 10)\) and \(w=x + 10\), then \(l = 4x-1\).

Step2: Check if it is a square

A square has \(l = w\). Here \(l=4x-1\) and \(w=x + 10\). \(4x-1=x + 10\) gives \(4x-x=10 + 1\), \(3x=11\), \(x=\frac{11}{3}\). It is not a square for all \(x\).

Step3: Check the perimeter formula

The perimeter \(P=2(l + w)\). Substitute \(l = 4x-1\) and \(w=x + 10\)

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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.