1. **State the problem:** Find the coefficient of $x$ in the expansion of $\left(5x - 3\right)\left(2x^2 - 3x + 1\right)$.\n\n2. **Write the expression to expand:** \n$$\left(5x - 3\right)\left(2x^2 - 3x + 1\right)$$\n\n3. **Use distributive property (FOIL) to expand:** \n$$= 5x \cdot 2x^2 + 5x \cdot (-3x) + 5x \cdot 1 - 3 \cdot 2x^2 - 3 \cdot (-3x) - 3 \cdot 1$$\n\n4. **Calculate each term:** \n$$= 10x^3 - 15x^2 + 5x - 6x^2 + 9x - 3$$\n\n5. **Combine like terms:** \n$$10x^3 + (-15x^2 - 6x^2) + (5x + 9x) - 3$$\n$$= 10x^3 - 21x^2 + 14x - 3$$\n\n6. **Identify the coefficient of $x$:** \nThe coefficient of $x$ is $14$.