Question

which expression is equal to \\((2x^3 + 9x^2 - 4x + 10) \div (x + 5)\\)?

1. \\(2x^2 - 4x + 9 - \frac{2}{x + 5}\\)
2. \\(2x^2 + 14x + 66 + \frac{340}{x + 5}\\)
3. \\(2x^2 + 4x - 24 - \frac{110}{x + 5}\\)
4. \\(2x^2 - x + 1 + \frac{5}{x + 5}\\)

Explanation:

Step1: Divide leading terms

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

Step2: Multiply divisor by $2x^2$

$2x^2(x+5) = 2x^3 +10x^2$

Step3: Subtract from dividend

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

Step4: Divide new leading terms

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

Step5: Multiply divisor by $-x$

$-x(x+5) = -x^2 -5x$

Step6: Subtract new remainder

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

Step7: Divide leading terms again

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

Step8: Multiply divisor by $1$

$1(x+5) = x+5$

Step9: Final remainder subtraction

$(x+10)-(x+5) = 5$

Answer:

1. $2x^2 - x +1+\frac{5}{x+5}$