1. **State the problem:** Solve the quadratic equation $$3x^2 + 7x - 10 = 0$$ completely. 2. **Recall the factoring method:** To factor a quadratic equation of the form $$ax^2 + bx + c = 0$$, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate product and sum:** Here, $$a = 3$$, $$b = 7$$, and $$c = -10$$. Calculate $$a \times c = 3 \times (-10) = -30$$. We need two numbers that multiply to $$-30$$ and add to $$7$$. 4. **Find the numbers:** The numbers are $$10$$ and $$-3$$ because $$10 \times (-3) = -30$$ and $$10 + (-3) = 7$$. 5. **Rewrite the middle term:** Rewrite $$7x$$ as $$10x - 3x$$: $$3x^2 + 10x - 3x - 10 = 0$$ 6. **Group terms:** Group the terms to factor by grouping: $$(3x^2 + 10x) - (3x + 10) = 0$$ 7. **Factor each group:** $$x(3x + 10) - 1(3x + 10) = 0$$ 8. **Factor out the common binomial:** $$(3x + 10)(x - 1) = 0$$ 9. **Solve each factor:** Set each factor equal to zero: $$3x + 10 = 0 \Rightarrow x = -\frac{10}{3}$$ $$x - 1 = 0 \Rightarrow x = 1$$ 10. **Final solution:** The solutions to the equation are $$x = 1$$ and $$x = -\frac{10}{3}$$.