1. **State the problem:** Solve for $m$ in the equation $$4^{m+2} + 4^{m+5} = 65.$$\n\n2. **Recall the properties of exponents:** For any base $a$ and exponents $x$ and $y$, $$a^{x+y} = a^x \cdot a^y.$$ Also, powers with the same base can be factored.\n\n3. **Rewrite the equation:**\n$$4^{m+2} + 4^{m+5} = 4^{m+2} + 4^{m+2+3} = 4^{m+2} + 4^{m+2} \cdot 4^3.$$\nSince $4^3 = 64$, the equation becomes\n$$4^{m+2} + 64 \cdot 4^{m+2} = 65.$$\n\n4. **Factor out $4^{m+2}$:**\n$$4^{m+2}(1 + 64) = 65,$$\nwhich simplifies to\n$$4^{m+2} \cdot 65 = 65.$$\n\n5. **Divide both sides by 65:**\n$$4^{m+2} = 1.$$\n\n6. **Solve for $m$:**\nSince $4^{m+2} = 1$, and $4^0 = 1$, it follows that\n$$m + 2 = 0,$$\nso\n$$m = -2.$$\n\n**Final answer:** $$m = -2.$$