1. **State the problem:** Solve algebraically for all values of $x$ in the equation $$2x^3 - 14x^2 + 2x - 14 = 0.$$\n\n2. **Factor out the greatest common factor (GCF):** Notice each term has a factor of 2, so factor it out:\n$$2(x^3 - 7x^2 + x - 7) = 0.$$\n\n3. **Divide both sides by 2:** Since 2 is nonzero, dividing both sides by 2 does not change the solutions:\n$$\cancel{2}(x^3 - 7x^2 + x - 7) = \cancel{2} \cdot 0 \Rightarrow x^3 - 7x^2 + x - 7 = 0.$$\n\n4. **Group terms to factor by grouping:**\nGroup as $(x^3 - 7x^2) + (x - 7)$:\n$$x^2(x - 7) + 1(x - 7) = 0.$$\n\n5. **Factor out the common binomial factor $(x - 7)$:**\n$$(x^2 + 1)(x - 7) = 0.$$\n\n6. **Set each factor equal to zero and solve:**\n- For $x - 7 = 0$, we get $x = 7$.\n- For $x^2 + 1 = 0$, solve for $x$:\n$$x^2 = -1 \Rightarrow x = \pm \sqrt{-1} = \pm i,$$ where $i$ is the imaginary unit.\n\n7. **Final solutions:**\n$$x = 7, \quad x = i, \quad x = -i.$$\n\nThese are all the roots of the cubic equation, including complex roots.