The rational expression $$ \frac{2 x+3}{x+4}+\frac{8}{x} $$ is equivalent to a.) $$ \frac{2 x^{2}+11 x+32}{x(x+4)} $$ b.) $$ \frac{2 x+11}{2 x+4} $$ c.) $$ \frac{2 x^{2}-5 x-32}{x(x+4)} $$ d.) $$ \frac{2 x-11}{2 x-4} $$

Question

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To determine which expression is equivalent to $$ \frac{2x + 3}{x + 4} + \frac{8}{x} $$, let's perform the addition step by step.

**Step 1: Find a Common Denominator**
The denominators are $$ x + 4 $$ and $$ x $$. The least common denominator (LCD) is $$ x(x + 4) $$.

**Step 2: Rewrite Each Fraction with the LCD**
$$
\frac{2x + 3}{x + 4} = \frac{(2x + 3) \cdot x}{x(x + 4)} = \frac{2x^2 + 3x}{x(x + 4)}
$$
$$
\frac{8}{x} = \frac{8 \cdot (x + 4)}{x(x + 4)} = \frac{8x + 32}{x(x + 4)}
$$

**Step 3: Add the Two Fractions**
$$
\frac{2x^2 + 3x}{x(x + 4)} + \frac{8x + 32}{x(x + 4)} = \frac{2x^2 + 3x + 8x + 32}{x(x + 4)} = \frac{2x^2 + 11x + 32}{x(x + 4)}
$$

**Conclusion:**
The simplified form of the given expression matches option **a.**

**Answer:** a.) $$ \frac{2 x^{2}+11 x+32}{x(x+4)} $$
The equivalent expression is $$ \frac{2x^{2}+11x+32}{x(x+4)} $$.

The rational expression is equivalent to
a.)
b.)
c.)
d.)

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