Answer
(a.)$ V(4)=128$.
The volume of the box is $128$ cubic inches whose height is $4$ inches.
(b.)$V(x)=x(x-2)(3x+4)$
(c.)$ V(4)=128$ and $V(5)=285$.
Work Step by Step
The given function is $V(x)=3x^3-2x^2-8x$.
(a.) Replace $x$ by $4$ into the given function.
$\Rightarrow V(4)=3(4)^3-2(4)^2-8(4)$
Clear the parentheses.
$\Rightarrow V(4)=192-32-32$
Add like terms.
$\Rightarrow V(4)=192-64$
$\Rightarrow V(4)=128$.
The volume of the box is $128$ cubic inches whose height is $4$ inches.
(b.) Factor $V(x)=3x^3-2x^2-8x$.
Factor out $x$ from all terms.
$\Rightarrow V(x)=x(3x^2-2x-8)$
Rewrite the middle term $-2x$ as $-6x+4x$.
$\Rightarrow V(x)=x(3x^2-6x+4x-8)$
Group the terms.
$\Rightarrow V(x)=x[(3x^2-6x)+(4x-8)]$
Factor each group.
$\Rightarrow V(x)=x[3x(x-2)+4(x-2)]$
Factor out $(x-2)$.
$\Rightarrow V(x)=x(x-2)(3x+4)$.
(c.) Replace $x$ by $4$ in the factored form.
$\Rightarrow V(4)=4(4-2)(3(4)+4)$
Simplify.
$\Rightarrow V(4)=4(2)(12+4)$
$\Rightarrow V(4)=4(2)(16)$
$\Rightarrow V(4)=128$
Replace $x$ by $5$ in the factored form.
$\Rightarrow V(5)=5(5-2)(3(5)+4)$
Simplify.
$\Rightarrow V(5)=5(3)(15+4)$
$\Rightarrow V(5)=5(3)(19)$
$\Rightarrow V(5)=285$.
