By the Remainder Theorem, the remainder when $p(x)$ is divided by $x-2$ is $p(2)$. Since that remainder is $5$, we know $p(2)=5$. Therefore, $(p(x)-5)$ has value $0$ when $x=2$, so $(x-2)$ must be a factor of $p(x)-5$. That means $p(x)-5$ is divisible by $x-2$.

To find when the stream is at least $12$ inches high, solve

First divide both sides by $-4$, which reverses the inequality:

Now expand:

Factor:

This inequality is true for values of $t$ between the roots, inclusive, so

The length of this time interval is $4-2=2$ seconds. Therefore, the correct answer is $2$.

Because $x-2$ is a factor of each polynomial, substituting $x=2$ must make each expression equal to $0$. For $x^2+ax-8$, we get $2^2+2a-8=0$, so $4+2a-8=0$, which simplifies to $2a-4=0$ and then $a=2$. For $x^2-4x+b$, we get $2^2-4(2)+b=0$, so $4-8+b=0$, which simplifies to $b=4$. Therefore, $a+b=2+4=6$.

To rewrite $n^3-4n$ in terms of $(n-2)$, let $u=n-2$. Then $n=u+2$. Substitute into the expression:

Expand:

so

Now replace $u$ with $(n-2)$:

So the equivalent expression is $(n-2{)}^3+6(n-2{)}^2+8(n-2)$.

For a polynomial written in factored form, each factor of the form $(x-r)$ gives an $x$-intercept at $x=r$. In $y=(x-2)(x+1)(x-k)$, the first two factors give intercepts at $x=2$ and $x=-1$, which match two of the points on the graph. The third factor, $(x-k)$, gives an intercept at $x=k$. Since the remaining intercept is at $(5,0)$, it follows that $k=5$.