Question

A rectangular prism has a length of 2x^2+2x-24 over 4x^2+x, a width of x^2+x-6 over x+4, and a height of 8x^2+2x over x^2-9. For all values of x for which it is defined , express, in terms of x, the volume of the prism in simplest form.
1. 4(x-2)
2. 2(x-2)
3.2(x+2)
4.4(x+2) please show work.

Answer 1

$V=abc\\\\ V=\frac{2x^2+2x-24}{4x^2+x}\cdot\frac{x^2+x-6}{x+4}\cdot\frac{8x^2+2x}{x^2-9}\\ V=\frac{2(x^2+x-12)}{x(4x+1)}\cdot\frac{x^2+3x-2x-6}{x+4}\cdot\frac{2x(4x+1)}{(x-3)(x+3)}\\ V=2(x^2+4x-3x-12)\cdot\frac{x(x+3)-2(x+3)}{x+4}\cdot\frac{2}{(x-3)(x+3)}\\ V=2(x(x+4)-3(x+4))\cdot\frac{(x-2)(x+3)}{x+4}\cdot\frac{2}{(x-3)(x+3)}\\ V=2(x-3)(x+4)\cdot\frac{x-2}{x+4}\cdot\frac{2}{x-3}\\ V=2\cdot(x-2)\cdot2\\ V=4(x-2)$