1. **Stating the problem:** We have a hexagon with external angles labeled as algebraic expressions involving $x$: $x+64$, $2x$, $3x-54$, $5x$, $4x$, $3x$, $2x+10$. We need to find the value of $x$. 2. **Formula and rule:** The sum of the external angles of any polygon is always $360^\circ$. 3. **Set up the equation:** Sum all external angles: $$ (x+64) + 2x + (3x - 54) + 5x + 4x + 3x + (2x + 10) = 360 $$ 4. **Simplify the equation:** Combine like terms: $$ x + 64 + 2x + 3x - 54 + 5x + 4x + 3x + 2x + 10 = 360 $$ $$ (x + 2x + 3x + 5x + 4x + 3x + 2x) + (64 - 54 + 10) = 360 $$ $$ 20x + 20 = 360 $$ 5. **Solve for $x$:** $$ 20x = 360 - 20 $$ $$ 20x = 340 $$ $$ x = \frac{340}{20} = 17 $$ 6. **Answer:** The value of $x$ is $17$.