1. **State the problem:** Factor the expression $$3x^2(x - 7) + 2x(x - 7)$$ completely. 2. **Identify the common factor:** Both terms contain the binomial factor $$(x - 7)$$. 3. **Factor out the common binomial:** $$3x^2(x - 7) + 2x(x - 7) = (x - 7)(3x^2 + 2x)$$ 4. **Factor the remaining polynomial:** The expression inside the parentheses is $$3x^2 + 2x$$. Factor out the common factor $$x$$: $$3x^2 + 2x = x(3x + 2)$$ 5. **Write the fully factored form:** $$ (x - 7) \times x \times (3x + 2) = x(x - 7)(3x + 2) $$ **Final answer:** $$x(x - 7)(3x + 2)$$