Question

divide using polynomial long division or synthetic division.\\((x^4 + 3x^3 - 4x^2 + 5x + 3) \div (x^2 + x + 4) = \square\\)

Explanation:

Step1: Divide leading terms

$\frac{x^4}{x^2}=x^2$
Multiply divisor by $x^2$: $x^2(x^2+x+4)=x^4+x^3+4x^2$
Subtract from dividend:
$(x^4+3x^3-4x^2+5x+3)-(x^4+x^3+4x^2)=2x^3-8x^2+5x+3$

Step2: Divide new leading terms

$\frac{2x^3}{x^2}=2x$
Multiply divisor by $2x$: $2x(x^2+x+4)=2x^3+2x^2+8x$
Subtract from current polynomial:
$(2x^3-8x^2+5x+3)-(2x^3+2x^2+8x)=-10x^2-3x+3$

Step3: Divide new leading terms

$\frac{-10x^2}{x^2}=-10$
Multiply divisor by $-10$: $-10(x^2+x+4)=-10x^2-10x-40$
Subtract from current polynomial:
$(-10x^2-3x+3)-(-10x^2-10x-40)=7x+43$

Answer:

$x^2+2x-10+\frac{7x+43}{x^2+x+4}$