divide the polynomials. the form of your answer should either be $p(x)$ or $p(x)+\frac{k}{x + 1}$ where…

Explanation:

Step1: Use polynomial long - division

Divide the leading term of the numerator $3x^{3}+x - 11$ by the leading term of the denominator $x + 1$. The leading term of the numerator is $3x^{3}$ and of the denominator is $x$. So, the first term of the quotient is $3x^{2}$ since $\frac{3x^{3}}{x}=3x^{2}$.

Step2: Multiply and subtract

Multiply $x + 1$ by $3x^{2}$ to get $3x^{3}+3x^{2}$, and subtract it from the numerator: $(3x^{3}+x - 11)-(3x^{3}+3x^{2})=-3x^{2}+x - 11$.

Step3: Repeat the process

Divide the leading term of $-3x^{2}+x - 11$ (which is $-3x^{2}$) by the leading term of $x + 1$ (which is $x$). The result is $-3x$. Multiply $x + 1$ by $-3x$ to get $-3x^{2}-3x$, and subtract it from $-3x^{2}+x - 11$: $(-3x^{2}+x - 11)-(-3x^{2}-3x)=4x - 11$.

Step4: One more time

Divide the leading term of $4x - 11$ (which is $4x$) by the leading term of $x + 1$ (which is $x$). The result is $4$. Multiply $x + 1$ by $4$ to get $4x+4$, and subtract it from $4x - 11$: $(4x - 11)-(4x + 4)=-15$.

Answer:

$3x^{2}-3x + 4-\frac{15}{x + 1}$