6.When a polynomial x4+x3−10x2−1 is divided by another polynomial, the quotient is x3−3x2+2x−8 and the...

Given
Polynomial (dividend) ${\displaystyle x^4+x^3-10x^2-1}$
Remainder is 31
Polynomial (quotient) is ${\displaystyle x^3-3x^2+2x-8}$
To find the other polynomial(divisor)
Using division Formula
$\text{Dividend}=\text{quotient}\times \text{divisor}+\text{remainder}$
Rewriting the formula
${\displaystyle \Rightarrow }$ $\text{Dividend}-\text{remainder}=\text{quotient}\times \text{divisor}$
${\displaystyle \Rightarrow }$ $\frac{\text{Dividend}-\text{remainder}}{\text{quotient}}=\text{divisor}$
Hence
$\text{divisor}=\frac{\text{Divident}-\text{remainder}}{\text{quatient}}$
Plugin the known values
Divisor ${\displaystyle =\frac{x^4+x^3-10x^2-1-31}{x^3-3x^2+2x-8}}$
${\displaystyle \Rightarrow }$ divisor ${\displaystyle =\frac{x^4+x^3-10x^2-1-32}{x^3-3x^2+2x-8}}$
Factorising numerator
${\displaystyle \Rightarrow }$ divisor $=\frac{x^3-3x^2+2x-8(x+4)}{x^3-3x^2+2x-8}$
Canceling ${\displaystyle x^3-3x^2+2x-8}$ from numerator and denominator
${\displaystyle \Rightarrow }$ divisor $=(x+4)$
Hence other polynomial (divisor) ${\displaystyle =x+4}$
Conclusion
Hence the other polynomial which gives quotient ${\displaystyle x^3-3x^2+2x-8}$ and remainder 31 on dividing polynomial${\displaystyle x^4-x^3-10x^2-1}$ is ${\displaystyle x+4.}$
$\text{Hence the other polinomial (divisor) is}x+4$