1. **State the problem:** Factor the polynomial $4x^3 - 26x^2 + 30x$ completely. 2. **Identify the common factor:** Each term has a factor of $2x$, so factor it out first. 3. **Factor out the common factor:** $$4x^3 - 26x^2 + 30x = 2x(\cancel{2}x^2 - \cancel{13}x + \cancel{15})$$ 4. **Factor the quadratic inside the parentheses:** We want to factor $2x^2 - 13x + 15$. 5. **Find two numbers that multiply to $2 \times 15 = 30$ and add to $-13$:** These numbers are $-10$ and $-3$. 6. **Rewrite the middle term:** $$2x^2 - 10x - 3x + 15$$ 7. **Group terms and factor each group:** $$= (2x^2 - 10x) + (-3x + 15)$$ $$= 2x(x - 5) - 3(x - 5)$$ 8. **Factor out the common binomial:** $$= (2x - 3)(x - 5)$$ 9. **Write the complete factorization:** $$4x^3 - 26x^2 + 30x = 2x(2x - 3)(x - 5)$$ **Final answer:** $2x(2x - 3)(x - 5)$