1. **State the problem:** We need to divide the expression $\frac{x^2 + 2xy - 8y^2}{x^2 + 2} \div (x^2 + 4xy)$. 2. **Rewrite division as multiplication:** Dividing by a term is the same as multiplying by its reciprocal. So, $$\frac{x^2 + 2xy - 8y^2}{x^2 + 2} \div (x^2 + 4xy) = \frac{x^2 + 2xy - 8y^2}{x^2 + 2} \times \frac{1}{x^2 + 4xy}$$ 3. **Factor the numerator of the first fraction:** The quadratic expression $x^2 + 2xy - 8y^2$ can be factored by looking for two numbers that multiply to $-8y^2$ and add to $2y$: $$x^2 + 2xy - 8y^2 = (x + 4y)(x - 2y)$$ 4. **Rewrite the expression with factored numerator:** $$\frac{(x + 4y)(x - 2y)}{x^2 + 2} \times \frac{1}{x^2 + 4xy}$$ 5. **Factor the denominator of the second fraction:** $$x^2 + 4xy = x(x + 4y)$$ 6. **Substitute the factored form:** $$\frac{(x + 4y)(x - 2y)}{x^2 + 2} \times \frac{1}{x(x + 4y)}$$ 7. **Cancel common factors:** The term $x + 4y$ appears in numerator and denominator, so cancel it: $$\frac{(x - 2y)}{x^2 + 2} \times \frac{1}{x} = \frac{x - 2y}{x(x^2 + 2)}$$ 8. **Final simplified expression:** $$\boxed{\frac{x - 2y}{x(x^2 + 2)}}$$ --- **Graph description:** The parabola opening upwards with vertex at the origin and symmetric x-intercepts corresponds to $y = ax^2$. The line crossing from bottom-left to top-right corresponds to $y = bx + c$. The division expression simplifies to a rational function involving $x$ and $y$ as above.