Simplify the complex fraction. $\frac { - \frac { 5 } { x } - \frac { 10 } { x + 5 } } { \frac { 3 } { x ^ { 2 } + 5 x } + \frac { 3 } { x } }$
A) $- \frac { 5 ( 3 x - 5 ) } { 3 ( x + 6 ) }$
B) $- \frac { 5 ( 3 x + 5 ) } { 3 ( x + 6 ) }$
C) $- \frac { 3 ( 3 x - 5 ) } { 5 ( x + 6 ) }$

Question

Quizplus · Accepted Answer

To simplify the complex fraction, we need to combine the fractions in the numerator and denominator.

First, let's combine the fractions in the numerator:

\begin{align*}
-\frac{5}{x}-\frac{10}{x+5}&=-\frac{5(x+5)}{x(x+5)}-\frac{10x}{x(x+5)}\
&=-\frac{5x+25-10x}{x(x+5)}\
&=-\frac{-5x+25}{x(x+5)}\
&=\frac{5(x-5)}{x(x+5)}
\end{align*}

Next, let's combine the fractions in the denominator:

\begin{align*}
\frac{3}{x^2+5x}+\frac{3}{x}&=\frac{3}{x(x+5)}+\frac{3(x+5)}{x(x+5)}\
&=\frac{3+3x+15}{x(x+5)}\
&=\frac{3x+18}{x(x+5)}\
&=\frac{3(x+6)}{x(x+5)}
\end{align*}

Now we can substitute the simplified numerator and denominator back into the original expression:

\begin{align*}
\frac{\frac{5(x-5)}{x(x+5)}}{\frac{3(x+6)}{x(x+5)}}&=\frac{5(x-5)}{3(x+6)}\cdot\frac{x(x+5)}{x(x+5)}\
&=-\frac{5(x-5)}{3(x+6)}
\end{align*}

Therefore, the answer is choice B.

Simplify the Complex Fraction −5x−10x+53x2+5x+3x\frac { - \frac { 5 } { x } - \frac { 10 } { x + 5 } } { \frac { 3 } { x ^ { 2 } + 5 x } + \frac { 3 } { x } }x2+5x3​+x3​−x5​−x+510​​

Simplify the Complex Fraction $\frac { - \frac { 5 } { x } - \frac { 10 } { x + 5 } } { \frac { 3 } { x ^ { 2 } + 5 x } + \frac { 3 } { x } }$