Question

1. $(3x^{4} + 4x^{3} - 2x^{2} + 3x - 2) div (x + 2)$

Explanation:

Step1: Set up polynomial long division

Divide $3x^4 + 4x^3 - 2x^2 + 3x - 2$ by $x+2$.
First term: $\frac{3x^4}{x}=3x^3$
Multiply divisor: $3x^3(x+2)=3x^4+6x^3$
Subtract from dividend:
$(3x^4 + 4x^3 - 2x^2 + 3x - 2)-(3x^4+6x^3)=-2x^3-2x^2+3x-2$

Step2: Next term of quotient

$\frac{-2x^3}{x}=-2x^2$
Multiply divisor: $-2x^2(x+2)=-2x^3-4x^2$
Subtract:
$(-2x^3-2x^2+3x-2)-(-2x^3-4x^2)=2x^2+3x-2$

Step3: Next term of quotient

$\frac{2x^2}{x}=2x$
Multiply divisor: $2x(x+2)=2x^2+4x$
Subtract:
$(2x^2+3x-2)-(2x^2+4x)=-x-2$

Step4: Final term of quotient

$\frac{-x}{x}=-1$
Multiply divisor: $-1(x+2)=-x-2$
Subtract:
$(-x-2)-(-x-2)=0$

Answer:

$3x^3 - 2x^2 + 2x - 1$