Question

A rectangular prism has a volume of 3x2 + 18x + 24. Its base has a length of x + 2 and a width of 3. Which expression represents the height of the prism?

Answer 1

Let base = \(b\)
height = \(h\)
width = \(w\)

\(\large \text{volume} = b\times h \times w\)\\
\(\large 3x^2+18x+24=\underbrace{(x+2)}_{b}\times h\times \underbrace{3}_{w}=3(x+2)h\)

So essentially you have this equation, and you need to find \(h\):
\( 3x^2+18x+24 = 3(x+2)h\)... how to solve for h? Divide both sides by \(3(x+2)\):

\[ \frac{3x^2+18x+24}{3(x+2)}=h\]

Now to simplify this, let's factor the numerator... you can first pull out a 3 easily, and then we will factor the rest of the polynomial:
\[ \frac{3(x^2+6x+8)}{3(x+2)}=h\] ... we will cancel out the 3 in the numerator and denominator:
\[ \frac{x^2+6x+8}{x+2}=h\], factor the numerator:
\[ \frac{(x+4)\cancel{(x+2)}}{\cancel{x+2}}=h\\
\, \\
\ h=x+4\]