Answer
See the explanation.
Work Step by Step
The remainder theorem states that when a polynomial p(x) is divided by a linear polynomial (x - a), then the remainder is equal to p(a). The remainder theorem enables us to calculate the remainder of the division of any polynomial by a linear polynomial, without actually carrying out the steps of the long division.
Note that the degree of the remainder polynomial is always 1 less than the degree of the divisor polynomial. Using this fact, when any polynomial is divided by a linear polynomial (whose degree is 1), the remainder must be a constant (whose degree is 0).
In our case, to find $f(-6)$ for $f(x)=x^4+7x^3+8x^2+11x+5$, we can use remainder theorem to by substituting $-6$ in places of $x$ and solving $f(x)$,
$f(-6)=(-6)^4+7(-6)^3+8(-6)^2+11(-6)+5=1296-1512+280-66+5=1581-1581=0$.
There we find the remainder of dividing $f(x)$ by $x+6$ easily using the remainder theorem instead of solving for the equation using long steps.
