1. We are asked to find the limit: $$\lim_{x \to 2} \frac{x^4 + 3x^3 - 10x^2}{x^2 - 2x}$$. 2. First, check if direct substitution is possible by plugging in $x=2$: $$\text{Numerator} = 2^4 + 3 \cdot 2^3 - 10 \cdot 2^2 = 16 + 24 - 40 = 0$$ $$\text{Denominator} = 2^2 - 2 \cdot 2 = 4 - 4 = 0$$ Since both numerator and denominator are zero, we have an indeterminate form $\frac{0}{0}$. 3. To resolve this, factor numerator and denominator to simplify the expression. 4. Factor denominator: $$x^2 - 2x = x(x - 2)$$ 5. Factor numerator $x^4 + 3x^3 - 10x^2$: First, factor out $x^2$: $$x^2(x^2 + 3x - 10)$$ Now factor the quadratic: $$x^2(x + 5)(x - 2)$$ 6. Substitute factored forms back into the limit expression: $$\lim_{x \to 2} \frac{x^2 (x + 5)(x - 2)}{x (x - 2)}$$ 7. Cancel common factor $(x - 2)$: $$\lim_{x \to 2} \frac{x^2 (x + 5)}{x} = \lim_{x \to 2} x (x + 5)$$ 8. Now substitute $x=2$: $$2 (2 + 5) = 2 \times 7 = 14$$ Final answer: $$\boxed{14}$$