Question

1. (2x⁴ + 5x³ + 9x² + 10x + 8) ÷ (x² + 4)

Explanation:

Step1: Divide leading terms

$\frac{2x^4}{x^2}=2x^2$

Step2: Multiply divisor by $2x^2$

$2x^2(x^2+4)=2x^4+8x^2$

Step3: Subtract from dividend

$(2x^4+5x^3+9x^2+10x+8)-(2x^4+8x^2)=5x^3+x^2+10x+8$

Step4: Divide new leading terms

$\frac{5x^3}{x^2}=5x$

Step5: Multiply divisor by $5x$

$5x(x^2+4)=5x^3+20x$

Step6: Subtract from new polynomial

$(5x^3+x^2+10x+8)-(5x^3+20x)=x^2-10x+8$

Step7: Divide new leading terms

$\frac{x^2}{x^2}=1$

Step8: Multiply divisor by 1

$1(x^2+4)=x^2+4$

Step9: Subtract to find remainder

$(x^2-10x+8)-(x^2+4)=-10x+4$

Answer:

Quotient: $2x^2+5x+1$, Remainder: $-10x+4$
Or written as: $2x^2+5x+1+\frac{-10x+4}{x^2+4}$