Minimum m + n for real roots

For real roots:

1. $m^2 - 8n \ge 0 \implies m^2 \ge 8n$
2. $4n^2 - 4m \ge 0 \implies n^2 \ge m$

Test small integer values for $n$:

• If $n = 1$: $m^2 \ge 8 \implies m \ge 3$, and $n^2 \ge m \implies 1 \ge m$ (impossible).
• If $n = 2$: $m^2 \ge 16 \implies m \ge 4$, and $n^2 \ge m \implies 4 \ge m$. Thus $m = 4$. $m + n = 4 + 2 = 6$.

Check if smaller sum is possible: $m+n=6$ works. Hence smallest possible $m + n = 6$.