1. **State the problem:** We need to find the limit $$\lim_{x\to 2} \frac{x^4 + 3x^3 - 10x^2}{x^2 - 2x}.$$\n\n2. **Check direct substitution:** Substitute $x=2$ into numerator and denominator: Numerator: $2^4 + 3\cdot 2^3 - 10 \cdot 2^2 = 16 + 24 - 40 = 0$ Denominator: $2^2 - 2 \cdot 2 = 4 - 4 = 0$ Since direct substitution gives $\frac{0}{0}$, an indeterminate form, we need to simplify the expression.\n\n3. **Factor numerator and denominator:** Denominator: $x^2 - 2x = x(x - 2)$ Numerator: Factor $x^2$ first: $$x^4 + 3x^3 - 10x^2 = x^2(x^2 + 3x - 10).$$ Now factor the quadratic: $$x^2 + 3x - 10 = (x + 5)(x - 2).$$ So numerator becomes: $$x^2 (x + 5)(x - 2).$$\n\n4. **Simplify the fraction:** $$\frac{x^2 (x + 5)(x - 2)}{x (x - 2)} = \frac{x^2 (x + 5) \cancel{(x - 2)}}{x \cancel{(x - 2)}} = \frac{x^2 (x + 5)}{x} = x (x + 5).$$\n\n5. **Evaluate the simplified limit:** $$\lim_{x \to 2} x (x + 5) = 2 (2 + 5) = 2 \times 7 = 14.$$\n\n**Final answer:** $$\boxed{14}.$$