Question

find the volume.
image of a rectangular prism with dimensions x + 4, 2x, and x - 3

Explanation:

Step1: Recall Volume Formula for Rectangular Prism

The volume \( V \) of a rectangular prism (box) is given by the product of its length, width, and height. So, \( V = l \times w \times h \). Here, the length is \( x + 4 \), the width is \( 2x \), and the height is \( x - 3 \).

Step2: Multiply the Expressions

First, multiply \( (x + 4) \) and \( 2x \):
\( (x + 4) \times 2x = 2x(x + 4) = 2x^2 + 8x \).

Then, multiply this result by \( (x - 3) \):
\( (2x^2 + 8x)(x - 3) \).
Using the distributive property (FOIL for polynomials):
\( 2x^2 \times x + 2x^2 \times (-3) + 8x \times x + 8x \times (-3) \)
\( = 2x^3 - 6x^2 + 8x^2 - 24x \).

Step3: Combine Like Terms

Combine the \( x^2 \) terms: \( -6x^2 + 8x^2 = 2x^2 \).
So, the volume is \( 2x^3 + 2x^2 - 24x \).

Answer:

The volume of the rectangular prism is \( \boldsymbol{2x^3 + 2x^2 - 24x} \).