1. **State the problem:** We need to find the value of $x$ that satisfies the equation $$\frac{x-3}{4} + \frac{2}{3} = \frac{17}{12}.$$\n\n2. **Write the formula and rules:** To solve for $x$, we will isolate $x$ by performing algebraic operations such as addition, subtraction, multiplication, and division. We will also find a common denominator to combine fractions.\n\n3. **Find a common denominator:** The denominators are 4, 3, and 12. The least common denominator (LCD) is 12.\n\n4. **Rewrite each term with denominator 12:**\n$$\frac{x-3}{4} = \frac{3(x-3)}{12}, \quad \frac{2}{3} = \frac{8}{12}.$$\n\n5. **Rewrite the equation:**\n$$\frac{3(x-3)}{12} + \frac{8}{12} = \frac{17}{12}.$$\n\n6. **Combine the fractions on the left:**\n$$\frac{3(x-3) + 8}{12} = \frac{17}{12}.$$\n\n7. **Since denominators are equal, set numerators equal:**\n$$3(x-3) + 8 = 17.$$\n\n8. **Expand and simplify:**\n$$3x - 9 + 8 = 17,$$\n$$3x - 1 = 17.$$\n\n9. **Add 1 to both sides:**\n$$3x - 1 + 1 = 17 + 1,$$\n$$3x = 18.$$\n\n10. **Divide both sides by 3:**\n$$x = \frac{18}{3}.$$\n$$x = 6.$$\n\n**Final answer:** $x = 6$.\n\nThis corresponds to choice 2.