1. **State the problem:** We have a circle with center C and two tangent points B and D. Tangent segments from points B and D to point A are given as expressions in terms of $x$: - Tangent segment from D to A: $2x + 5$ - Tangent segment from B to A: $3x^2 + 2x - 7$ Since both are tangent segments from the same external point A to the circle, their lengths must be equal. 2. **Set up the equation:** $$2x + 5 = 3x^2 + 2x - 7$$ 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: $$x = \pm \sqrt{4}$$ $$x = \pm 2$$ **Final answer:** $$x = 2 \text{ or } x = -2$$