1. **State the problem:** We have two tangent segments from an external point A to a circle, touching the circle at points B and D. The lengths of these tangent segments are given as $2x + 5$ and $3x^2 + 2x - 7$. We need to find the values of $x$. 2. **Key property:** Tangent segments from a common external point to a circle are equal in length. Therefore, we set the two expressions equal: $$2x + 5 = 3x^2 + 2x - 7$$ 3. **Simplify the equation:** Subtract $2x$ from both sides: $$\cancel{2x} + 5 = 3x^2 + \cancel{2x} - 7$$ which simplifies to $$5 = 3x^2 - 7$$ 4. **Isolate the quadratic term:** Add 7 to both sides: $$5 + 7 = 3x^2$$ $$12 = 3x^2$$ 5. **Divide both sides by 3:** $$\frac{12}{3} = \frac{3x^2}{3}$$ which simplifies to $$4 = x^2$$ 6. **Solve for $x$:** Take the square root of both sides: $$x = \pm \sqrt{4}$$ $$x = \pm 2$$ 7. **Final answer:** The values of $x$ are $$\boxed{\pm 2}$$