Question

1. \\((x^4 - 6x^2 - 5) \div (x + 2)\\)

Explanation:

Step1: Write polynomial in standard form

$x^4 + 0x^3 - 6x^2 + 0x - 5$

Step2: Use synthetic division with root $x=-2$

Set up synthetic division table:

$$\begin{array}{r|rrrrr} -2 & 1 & 0 & -6 & 0 & -5 \\ & & -2 & 4 & 4 & -8 \\ \hline & 1 & -2 & -2 & 4 & -13 \\ \end{array}$$

Step3: Identify quotient and remainder

Quotient: $x^3 - 2x^2 - 2x + 4$, Remainder: $-13$

Step4: Express as quotient + remainder/divisor

$\text{Result} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$

Answer:

$x^3 - 2x^2 - 2x + 4 - \frac{13}{x+2}$