1. **State the problem:** Solve the quadratic equation $$3x^2 + 7x - 10 = 0$$ completely. 2. **Formula and rules:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, we can use factoring if possible, or the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Attempt factoring:** We look for two numbers that multiply to $$a \times c = 3 \times (-10) = -30$$ and add to $$b = 7$$. 4. The numbers 10 and -3 satisfy this because $$10 \times (-3) = -30$$ and $$10 + (-3) = 7$$. 5. Rewrite the middle term using these numbers: $$3x^2 + 10x - 3x - 10 = 0$$ 6. Factor by grouping: $$ (3x^2 + 10x) - (3x + 10) = 0$$ $$ x(3x + 10) - 1(3x + 10) = 0$$ 7. Factor out the common binomial: $$ (x - 1)(3x + 10) = 0$$ 8. Set each factor equal to zero and solve: $$x - 1 = 0 \Rightarrow x = 1$$ $$3x + 10 = 0 \Rightarrow 3x = -10 \Rightarrow x = -\frac{10}{3}$$ **Final answer:** $$x = 1$$ or $$x = -\frac{10}{3}$$