Let's solve this step by step! 🎉 1. Imagine you have $x + \frac{1}{x} = 99$. 2. We want to find $\frac{100x}{2x^2 + 102x + 2}$. 3. First, notice the denominator: $2x^2 + 102x + 2$. 4. Let's try to rewrite the denominator: $$2x^2 + 102x + 2 = 2(x^2 + 51x + 1)$$ 5. Now, remember $x + \frac{1}{x} = 99$. Let's find $x^2 + \frac{1}{x^2}$: $$\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} = 99^2 = 9801$$ 6. So, $$x^2 + \frac{1}{x^2} = 9801 - 2 = 9799$$ 7. Multiply both sides by $x^2$: $$x^2 + 51x + 1 = ?$$ We want to express $x^2 + 51x + 1$ in terms of $x + \frac{1}{x}$. 8. Notice $x^2 + 51x + 1$ is not directly related, but let's split the denominator expression: $$2x^2 + 102x + 2 = 2(x^2 + 51x + 1)$$ Try to write numerator and denominator in terms of $x + \frac{1}{x}$. 9. Let's rewrite denominator: $$2x^2 + 102x + 2 = 2x^2 + 102x + 2 = 2x^2 + 2 + 102x$$ 10. Rewrite numerator: $$100x$$ 11. Divide numerator and denominator by $x$: $$\frac{100x}{2x^2 + 102x + 2} = \frac{100}{2x + 102 + \frac{2}{x}}$$ 12. Now, $x + \frac{1}{x} = 99$ so multiply by 2: $$2x + \frac{2}{x} = 2 \times 99 = 198$$ 13. So denominator becomes: $$2x + 102 + \frac{2}{x} = 198 + 102 = 300$$ 14. So the whole expression is: $$\frac{100}{300} = \frac{1}{3}$$ Great job! 🎯 **Final answer:** $$\boxed{\frac{1}{3}}$$