1. **State the problem:** Solve for $x$ in the equation $$\log_2(x^2 + 7) - \log_2(x + 5) = 3.$$\n\n2. **Recall the logarithm subtraction rule:** $$\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right).$$\nApplying this, the equation becomes $$\log_2 \left(\frac{x^2 + 7}{x + 5}\right) = 3.$$\n\n3. **Rewrite the logarithmic equation in exponential form:** $$\frac{x^2 + 7}{x + 5} = 2^3 = 8.$$\n\n4. **Solve the resulting rational equation:** Multiply both sides by $x + 5$ (noting $x \neq -5$ to avoid division by zero):\n$$x^2 + 7 = 8(x + 5).$$\n\n5. **Expand and simplify:**\n$$x^2 + 7 = 8x + 40.$$\n\n6. **Bring all terms to one side:**\n$$x^2 - 8x + 7 - 40 = 0 \implies x^2 - 8x - 33 = 0.$$\n\n7. **Solve the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $a=1$, $b=-8$, $c=-33$.\nCalculate the discriminant:\n$$\Delta = (-8)^2 - 4 \times 1 \times (-33) = 64 + 132 = 196.$$\n\n8. **Find the roots:**\n$$x = \frac{8 \pm \sqrt{196}}{2} = \frac{8 \pm 14}{2}.$$\n\n9. **Calculate each root:**\n- $$x = \frac{8 + 14}{2} = \frac{22}{2} = 11,$$\n- $$x = \frac{8 - 14}{2} = \frac{-6}{2} = -3.$$\n\n10. **Check for extraneous solutions:** The domain requires $x + 5 > 0 \Rightarrow x > -5$ and $x^2 + 7 > 0$ always true. Both $x=11$ and $x=-3$ satisfy $x > -5$.\n\n**Final answer:** $$\boxed{x = 11 \text{ or } x = -3}.$$