divide the polynomials. the form of your answer should either be $p(x)$ or $p(x)+\frac{k}{x - 2}$ where…

Explanation:

Step1: Use polynomial long - division

We divide $x^{3}+6x^{2}-5x$ by $x - 2$. First, divide the leading term of the dividend $x^{3}$ by the leading term of the divisor $x$. The result is $x^{2}$. Multiply $x - 2$ by $x^{2}$ to get $x^{3}-2x^{2}$. Subtract this from the dividend: $(x^{3}+6x^{2}-5x)-(x^{3}-2x^{2})=8x^{2}-5x$.

Step2: Continue the division

Divide the leading term of the new dividend $8x^{2}$ by the leading term of the divisor $x$, we get $8x$. Multiply $x - 2$ by $8x$ to get $8x^{2}-16x$. Subtract this from $8x^{2}-5x$: $(8x^{2}-5x)-(8x^{2}-16x)=11x$.

Step3: Final division step

Divide the leading term of the new dividend $11x$ by the leading term of the divisor $x$, we get $11$. Multiply $x - 2$ by $11$ to get $11x-22$. Subtract this from $11x$: $11x-(11x - 22)=22$. So, $\frac{x^{3}+6x^{2}-5x}{x - 2}=x^{2}+8x + 11+\frac{22}{x - 2}$.

Answer:

$x^{2}+8x + 11+\frac{22}{x - 2}$