Question

what is the base area of box 3?
$x^2 + \boxed{\phantom{4}}x$
complete
what is the volume of box 3?
$4x^3 + 2x^2 + 4x + 1$
$x^4 + 4x^3 + x^2 + 4x$
$x^4 + 4x^3 + 2x^2 + 4x$
$x^4 + 2x^3 + x^2 + 4$
done

Explanation:

Step1: Identify base area

The base area of Box 3 is given as $x^2 + 4x$.

Step2: Assume height of Box 3

For a rectangular box, volume = base area × height. Observing the polynomial options, the height is $x^2 + 2x$ (to match the degree 4 polynomial result).

Step3: Calculate volume

Multiply base area by height:
$$(x^2 + 4x)(x^2 + 2x)$$
Expand using distributive property:
$$x^2(x^2 + 2x) + 4x(x^2 + 2x)$$
$$=x^4 + 2x^3 + 4x^3 + 8x^2? \text{ Correction: Recheck standard expansion for matching options.}$$
Correct expansion for target option:
$$(x^2 + 4x)(x^2 + 2x) = x^4 + 2x^3 + 4x^3 + 8x^2? \text{ No, match the given options:}$$
Wait, re-express base area as $x^2 + 4x$, multiply by $x^2 + 2x$:
$$x^2 \cdot x^2 + x^2 \cdot 2x + 4x \cdot x^2 + 4x \cdot 2x$$
$$=x^4 + 2x^3 + 4x^3 + 8x^2? \text{ No, correct matching:}$$
Wait, the correct expansion for the option $x^4 + 4x^3 + 2x^2 + 4x$ is $(x^2 + 4x)(x^2 + 0x + 1)$? No, recheck:
Wait, base area is $x^2 + 4x$, if height is $x^2 + 2x + 1$? No, the correct match is:
$$(x^2 + 4x)(x^2 + 2x + 1) = x^4 + 2x^3 + x^2 + 4x^3 + 8x^2 + 4x = x^4 + 6x^3 + 9x^2 + 4x \text{ No.}$$
Wait, the correct option is $x^4 + 4x^3 + 2x^2 + 4x$, factor it:
$$x(x^3 + 4x^2 + 2x + 4) = x[x^2(x+4) + 2(x+4)] = x(x^2+2)(x+4) = (x^2+4x)(x^2+2)$$
Ah, so height is $x^2 + 2$. Multiply base area by height:
$$(x^2 + 4x)(x^2 + 2)$$
$$=x^2(x^2+2) + 4x(x^2+2)$$
$$=x^4 + 2x^2 + 4x^3 + 8x? \text{ No, correction:}$$
$$x^2 \cdot x^2 = x^4, x^2 \cdot 2 = 2x^2, 4x \cdot x^2 = 4x^3, 4x \cdot 2 = 8x \text{ No, mismatch.}$$
Wait, correct expansion for $x^4 + 4x^3 + 2x^2 + 4x$:
$$x^4 + 4x^3 + 2x^2 + 4x = x^3(x+4) + 2x(x+2) \text{ No, factor as } (x^2 + 4x)(x^2 + 0x + 1) + 2x^2? \text{ No, the correct step is:}$$
Given base area $A = x^2 + 4x$, volume $V = A \times h$. The only option that is a multiple of $x(x+4)$ (factored base area) is $x^4 + 4x^3 + 2x^2 + 4x = x(x^3 + 4x^2 + 2x + 4) = x[x^2(x+4) + 2(x+2)]$? No, $x^4 + 4x^3 + 2x^2 + 4x = x^2(x^2+4x) + 2(x^2+2x)$? No, the correct match is that when we multiply $(x^2 + 4x)$ by $(x^2 + 2)$ we get $x^4 + 2x^2 + 4x^3 + 8x$, which is not an option. Wait, the base area is $x^2 + 4x$, the correct volume option is $x^4 + 4x^3 + 2x^2 + 4x$, which factors to $x(x^3 + 4x^2 + 2x + 4) = x(x+4)(x^2 + 0.5x + 1)$? No, I made a mistake:
Wait, $x^4 + 4x^3 + 2x^2 + 4x = x(x^3 + 4x^2 + 2x + 4) = x[ x^2(x+4) + 2(x+2) ]$ is wrong. Correct factoring: $x^3 + 4x^2 + 2x + 4 = x^2(x+4) + 2(x+2)$ no, $x^3 +4x^2 +2x +4 = (x^3 +4x^2) + (2x+4) = x^2(x+4) + 2(x+2)$ no, but if base area is $x^2 + 4x$, then volume is $(x^2+4x)(x^2 + a x + b) = x^4 + (a+4)x^3 + (b+4a)x^2 +4b x$. Match with $x^4 +4x^3 +2x^2 +4x$:

1. $a+4=4 \implies a=0$
2. $b+4a=2 \implies b=2$
3. $4b=4 \implies b=1$, contradiction. Wait, the first question's base area is $1x^2 +4x$, so $x^2+4x$. Then the correct volume is $(x^2+4x)(x^2 + 2x + 1)$ is wrong. Wait, the option $x^4 +4x^3 +2x^2 +4x$ is the only one that has $4x$ as the constant term, which matches $4x \times 1$, so height is $x^2 + 0x +1$, then volume is $(x^2+4x)(x^2+1) = x^4 +x^2 +4x^3 +4x = x^4 +4x^3 +x^2 +4x$, which is the second option. Oh! I misread the base area: the base area is $x^2 +4x$, so $(x^2+4x)(x^2+1) = x^4 +x^2 +4x^3 +4x = x^4 +4x^3 +x^2 +4x$.

Step4: Confirm correct expansion

$$(x^2 + 4x)(x^2 + 1) = x^2 \cdot x^2 + x^2 \cdot 1 + 4x \cdot x^2 + 4x \cdot 1$$
$$=x^4 + x^2 + 4x^3 + 4x = x^4 + 4x^3 + x^2 + 4x$$

Answer:

$\boldsymbol{O.\ x^4 + 4x^3 + x^2 + 4x}$