box 3: length = 4 + x width = x height = x^2 + 1 the polynomial that represents the volume of box 3 has a…

Explanation:

Step1: Recall volume formula

The volume $V$ of a box is $V = \text{Length}\times\text{Width}\times\text{Height}$. Given $\text{Length}=4 + x$, $\text{Width}=x$, and $\text{Height}=x^{2}+1$. So $V=(4 + x)\times x\times(x^{2}+1)$.

Step2: First - multiply $(4 + x)$ and $x$

Using the distributive property $(a + b)c=ac+bc$, we have $(4 + x)x=4x+x^{2}$.

Step3: Then multiply the result by $(x^{2}+1)$

$(4x + x^{2})(x^{2}+1)=4x\times x^{2}+4x\times1+x^{2}\times x^{2}+x^{2}\times1$.

Step4: Simplify the products

$4x\times x^{2}=4x^{3}$, $4x\times1 = 4x$, $x^{2}\times x^{2}=x^{4}$, $x^{2}\times1=x^{2}$. So $V = x^{4}+4x^{3}+x^{2}+4x$.

Step5: Determine the degree

The degree of a polynomial is the highest power of the variable. Here, the highest - power of $x$ is 4.

Answer:

4