Question

p(x) = $x^3+4x^2-5x+6$
g(x) = x + 1
and verify with $p\left(x\right)\left[g\right(x)\times q(x\left)\right]+r\left(x\right)$

Answer 1

considering the question to be:

To prove $p\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$

$\frac{x^3+4x^2-5x+6}{x+1}$

$=x^2+\frac{3x^2-5x+6}{x+1}$

$=x^2+3x+\frac{-8x+6}{x+1}$

$=x^2+3x-8+\frac{14}{x+1}$

$p\left(x\right)=x^3+4x^2-5x+6,g\left(x\right)=x+1,q\left(x\right)=x^2+3x-8,r\left(x\right)=14$

To prove

$p\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$

$g\left(x\right)q\left(x\right)+r\left(x\right)$

$=(x+1)(x^2+3x-8)+14$

$=x^3+3x^2-8x+x^2+3x-8+14$

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

$=p\left(x\right)$