Simplify the expression completely if possible. $$ \frac{8 x}{x^{3}+4 x^{2}} $$

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UpStudy AI · Accepted Answer

To simplify the expression

$$
\frac{8x}{x^3 + 4x^2},
$$

we start by factoring the denominator. The expression in the denominator is

$$
x^3 + 4x^2.
$$

We can factor out the common term $$x^2$$:

$$
x^3 + 4x^2 = x^2(x + 4).
$$

Now, we can rewrite the original expression with the factored denominator:

$$
\frac{8x}{x^2(x + 4)}.
$$

Next, we can simplify the fraction by canceling out common factors in the numerator and the denominator. The numerator $$8x$$ can be expressed as $$8 \cdot x$$, and we see that there is an $$x$$ in the numerator and $$x^2$$ in the denominator. We can cancel one $$x$$ from the numerator with one $$x$$ from $$x^2$$:

$$
\frac{8x}{x^2(x + 4)} = \frac{8}{x(x + 4)}.
$$

Thus, the simplified expression is

$$
\frac{8}{x(x + 4)}.
$$

This is the final simplified form of the expression. Therefore, the answer is

$$
\boxed{\frac{8}{x(x + 4)}}.
$$
The simplified expression is $$\frac{8}{x(x + 4)}$$.

Simplify the expression completely if possible. \[ \frac{8 x}{x^{3}+4 x^{2}} \]

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