Use the factor theorem to determine if the given binomial is a factor of f(x). f(x)=x^4+8x^3+11x^2-11x+3, x+3

Step 1
To decide if the given binomial divides the polynomial f(x), using the factor theorem
Step 2
Recall the factor theorem: $(x-a)$ is a factor of the polynomial f(x) if and only if $f(a)=0$. Apply this in the present case with $a=-3$, that is , test whether f(-3) is zero or otherwise.
${\displaystyle f\left(x\right)=x^4+8x^3+11x^2-11x+3}$.
$f(-3)=81-216+99+33+3=0$
So, $(x+3)$ is a factor of f(x)
Step 3
Result: $x+3$ is a factor of f(x) (see factorization above)
${\displaystyle f\left(x\right)=x^4+8x^3+11x^2-11x+3}$
${\displaystyle =(x+3)(x^3+5x^2-4x+1)}$