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The width,w, of a rectangular garden is X -2 the area of the garden is X^3-2X-4 what is an expression for the length of the garden?

AX^2-2x-2
BX^2+2x-2
CX^2-2x+2
DX^2+2x+2

Verified Answer :

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\(A=lw \Rightarrow l=\frac{A}{w}\)
A - area, l - length, w - width

\(w=x-2 \\ A=x^3-2x-4 \\ \\ l=\frac{x^3-2x-4}{x-2}=\frac{x^3-2x^2+2x^2-4x+2x-4}{x-2}=\frac{x^2(x-2)+2x(x-2)+2(x-2)}{x-2}= \\ =\frac{(x^2+2x+2)(x-2)}{x-2}=x^2+2x+2\)

The answer is D. x²+2x+2.

Answer:

D. \(x^2+2x+2\)

Step-by-step explanation:

We know that,

The area of a rectangle is,

A = l × b,

Where, l is the length of the rectangle,

w is the width of the rectangle,

Given,

\(A = x^3-2x-4\)

\(w=(x-2)\)

By substituting values,

\(x^3-2x-4=(x-2)l\)

\(\implies l = \frac{x^3-2x-4}{x-2}=x^2+2x+2\) ( By long division shown below )

Hence, the length of the rectangular garden is \(x^2+2x+2\)

Option D is correct.

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