Perform the operation indicated. Simplify. $$ \frac{x+2 y}{24 x^{2}} \cdot \frac{3 x}{x^{2}+x y-2 y^{2}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{x + 2y}{24x^2} \cdot \frac{3x}{x^2 + xy - 2y^2}
$$

**Step 1: Factor the Denominator**

First, factor the quadratic expression in the denominator:

$$
x^2 + xy - 2y^2
$$

This factors as:

$$
(x + 2y)(x - y)
$$

**Step 2: Rewrite the Expression**

Substitute the factored form into the original expression:

$$
\frac{x + 2y}{24x^2} \cdot \frac{3x}{(x + 2y)(x - y)}
$$

**Step 3: Cancel Common Factors**

Notice that $$(x + 2y)$$ appears in both the numerator and the denominator, so they cancel out:

$$
\frac{3x}{24x^2(x - y)}
$$

**Step 4: Simplify the Coefficients and Variables**

Simplify the coefficients and the variables:

$$
\frac{3x}{24x^2(x - y)} = \frac{3}{24x(x - y)} = \frac{1}{8x(x - y)}
$$

**Final Answer:**

$$
\boxed{\dfrac{1}{8\,x\,\left(x - y\,\right)}}
$$
$$ \frac{1}{8x(x - y)} $$

Perform the operation indicated. Simplify. \( \frac{x+2 y}{24 x^{2}} \cdot \frac{3 x}{x^{2}+x y-2 y^{2}} \)