1. **State the problem:** Find the discriminant of the quadratic expression derived from $5^2 - 4(q)(-q + 3)$, which simplifies to $4q^2 - 12q + 25$. 2. **Recall the discriminant formula:** For a quadratic equation $ax^2 + bx + c$, the discriminant $\Delta$ is given by: $$\Delta = b^2 - 4ac$$ 3. Identify coefficients from $4q^2 - 12q + 25$: $$a = 4, \quad b = -12, \quad c = 25$$ 4. Substitute values into the discriminant formula: $$\Delta = (-12)^2 - 4 \times 4 \times 25$$ 5. Calculate each term: $$(-12)^2 = 144$$ $$4 \times 4 \times 25 = 400$$ 6. Compute the discriminant: $$\Delta = 144 - 400 = -256$$ 7. **Interpretation:** Since $\Delta < 0$, the quadratic has no real roots; it has two complex roots. **Final answer:** $$\boxed{-256}$$