How do you find the length, width, and height of a rectangular prism if the volume is \[{{h}^{3}}+{{h}^{2}}-20h\] cubic meters?

Hint: A polynomial is factored completely when it is expressed as a product of one or more polynomials that cannot be factored further. To factor a polynomial completely, we need to identify the greatest common monomial factor. Not all polynomials can be factored in. We know that the volume of a rectangular prism is \[V=l.b.h\].

Complete step by step answer:
As per the given question, we have to find the dimensions of the rectangular prism using factoring methods by factoring the given volume expression. Here, we have the given volume expression \[V={{h}^{3}}+{{h}^{2}}-20h\].

Let a rectangular prism with the dimension’s length ‘l’, width ‘b’ and height ‘h’ as shown in the figure below:


Then, the volume of the prism is given by \[V=l.b.h\]. But we know that the volume of the required prism is \[V={{h}^{3}}+{{h}^{2}}-20h\]. Hence, we can combine both expressions. Then, we get
\[\Rightarrow l.b.h={{h}^{3}}+{{h}^{2}}-20h\].
Here, we can observe that ‘h’ is common on both sides of the equation. Thus, we can eliminate ‘h’ to get
\[\Rightarrow \dfrac{l.b.h}{h}=\dfrac{{{h}^{3}}+{{h}^{2}}-20h}{h}\Rightarrow l.b={{h}^{2}}+h-20\]
In the quadratic equation \[{{h}^{2}}+h-20\], x-coefficient is 1. The product of \[{{x}^{2}}-\text{coefficient}\] and the constant term is -20. We split up x-coefficient 1 into two numbers whose sum (or difference) is 1 and product is -20. Hence, the required numbers are 5 and -4. Thus, the equation becomes
\[\Rightarrow l.b={{h}^{2}}+h-20={{h}^{2}}+5h-4h-20\]
Taking \[(h+5)\] common in the first 2 terms and last 2 terms, we get
\[\Rightarrow l.b=h(h+5)-4(h+5)=(h-4)(h+5)\]
As we know that length is greater than width, then we get \[l=(h+5)\] meters and \[b=(h-4)\] meters.
\[\therefore l=(h+5),\text{ }b=(h-4)\text{ and }h=h\] meters are the length, width and height of the rectangular prism respectively.

Note:
 In order to solve these types of questions, we need to have enough knowledge of factoring methods of polynomials. If polynomials can’t be factored then we can use quadratic formula \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] to find the factors. We should avoid calculation mistakes to get the correct solution.