1. **State the problem:** Factor the quadratic expression $7x^2 + 14x - 105$ and find the values of $x$ that satisfy $7x^2 + 14x - 105 = 0$. 2. **Write the equation:** $$7x^2 + 14x - 105 = 0$$ 3. **Factor out the greatest common factor (GCF):** The GCF of $7x^2$, $14x$, and $-105$ is 7. $$7\cancel{x^2} + 7\times 2x - 7\times 15 = 7(x^2 + 2x - 15) = 0$$ 4. **Divide both sides by 7:** $$\cancel{7}(x^2 + 2x - 15) = \cancel{7} \times 0 \Rightarrow x^2 + 2x - 15 = 0$$ 5. **Factor the quadratic $x^2 + 2x - 15$:** We look for two numbers that multiply to $-15$ and add to $2$. These numbers are $5$ and $-3$. $$x^2 + 2x - 15 = (x + 5)(x - 3) = 0$$ 6. **Solve for $x$:** Set each factor equal to zero: $$x + 5 = 0 \Rightarrow x = -5$$ $$x - 3 = 0 \Rightarrow x = 3$$ 7. **Final answer:** $$x = -5 \text{ or } x = 3$$ This corresponds to option C.