1. **State the problem:** Solve the equation $2^{x+2} - 2^{x+5} = -7$ for $x$. 2. **Recall the properties of exponents:** - $a^{m+n} = a^m \cdot a^n$ - We can factor expressions with common bases. 3. **Rewrite the terms:** $$2^{x+2} = 2^x \cdot 2^2 = 4 \cdot 2^x$$ $$2^{x+5} = 2^x \cdot 2^5 = 32 \cdot 2^x$$ 4. **Substitute back into the equation:** $$4 \cdot 2^x - 32 \cdot 2^x = -7$$ 5. **Factor out $2^x$:** $$2^x (4 - 32) = -7$$ $$2^x \cancel{(4 - 32)} = -7$$ $$2^x (-28) = -7$$ 6. **Divide both sides by $-28$:** $$2^x = \frac{-7}{-28} = \frac{7}{28}$$ $$2^x = \frac{1}{4}$$ 7. **Rewrite $\frac{1}{4}$ as a power of 2:** $$\frac{1}{4} = 2^{-2}$$ 8. **Set the exponents equal:** $$x = -2$$ **Final answer:** $$x = -2$$