Find the length and width of rectangle when you are given the area

Question

The area of a rectangle is $x^2 + 4x - 12$. What is the length and width of the rectangle?

The solution says the main idea is to factor $x^2 + 4x -12$.

So, since $-12 = -2 \times 6$ and $-2 + 6 = 4$, it can be written as $x^2 + 4x - 12 = (x - 2)(x + 6)$ since the length is usually the longer value, the length is $6$ and the width is $-2$.

I don't understand the logic to this solution at all. I understand $\text{length} \times \text{width} = \text{area}$, but outside of this information I don't understand how they got to this solution from the given information in the problem.

Answer 1

That is incorrect. Many rectangles, with different lengths and widths, can have the same area. Example: $2\times3 = 1\times6 = \pi\times\frac6\pi$

Also, the area is a function of $x$; if $x$ is not given, then the area is not given!

And width can't be negative.

Where did you find this "solution"? It's very wrong.

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EDIT1

Perhaps they just wanted you to factor the expression. Then the "answer" would have the length $(x+6)$ , and the width $(x-2)$ . But even this isn't unique; it could be $(2x+12)$ and $(\frac12 x-1)$ . Someone gave you a bad question.

Answer 2

The length is $(x+6)$ and the width is $(x-2)$. The area is $(x+6)(x-2)=x^2+4x-12$

You may take this answer as the length is 6 units longer and the width is 2 units shorter than a give number $x$

For example, given $x=10$, you may get that the area is $10^2+4\times10-12=128$, or, the area is $(10+6)(10-2)=16\times8=128$