Question

select the expression that is equivalent to \\(\frac{2x^3 + 11x^2 - 21x}{x^2 + 3x}\\) for \\(x \
eq -3\\) or \\(0\\) use long division*
a. \\(2x + 5 - \frac{6}{x + 3}\\)
b. \\(2x + 17 - \frac{20}{x + 3}\\)
c. \\(2x + 17 - \frac{36}{x^2 + 3x}\\)
d. \\(2x + 5 - \frac{36}{x + 3}\\)

Explanation:

Step1: Simplify numerator and denominator

First, factor out common terms:
Numerator: $2x^3 + 11x^2 - 21x = x(2x^2 + 11x - 21)$
Denominator: $x^2 + 3x = x(x + 3)$
Cancel $x$ (since $x
eq 0$):
$\frac{2x^2 + 11x - 21}{x + 3}$

Step2: Divide $2x^2+11x-21$ by $x+3$

Divide leading terms: $\frac{2x^2}{x} = 2x$. Multiply $x+3$ by $2x$: $2x^2 + 6x$.
Subtract from dividend:
$(2x^2 + 11x - 21) - (2x^2 + 6x) = 5x - 21$

Step3: Continue dividing the remainder

Divide leading term of remainder: $\frac{5x}{x} = 5$. Multiply $x+3$ by $5$: $5x + 15$.
Subtract from remainder:
$(5x - 21) - (5x + 15) = -36$

Step4: Write the final expression

Combine quotient and remainder over divisor:
$2x + 5 - \frac{36}{x + 3}$

Answer:

D. $2x + 5 - \frac{36}{x+3}$