$ \Delta = 3^2 - 4m = 9 - 4m \\\Delta \ge 0 \Rightarrow 9 - 4m \ge 0 \Leftrightarrow m \le \frac{9}{4} \\ Viet: x_1 + x_2 = -3; x_1x_2 = m \\ x_1^2 + x_2^2 = 17 \\\Leftrightarrow (x_1 + x_2)^2 - 2x_1x_2 = 17 \\\Leftrightarrow 3^2 - 2m = 17 \\\Leftrightarrow -2m = 8 \\\Leftrightarrow m = -4 $