Find the modulus of the sum of the roots of the equation x ^ 2 + 6x + 8 + | x + 4 | = 0x ^ 2 + 6x + 8 + | x + 4 | = 0.

Expanding the module, we get 2 equations:

x ^ 2 + 6x + 8 + x + 4 = 0 and x ^ 2 + 6x + 8 – x – 4 = 0.

Find the roots of the equations:

x ^ 2 + 7x + 12 = 0;

x12 = (-7 + – √49 – 4 * 12) / 2;

x1 = (-7 + 1) / 2 = -3; x2 = (- 8 – 1) = -4;

| x1 + x2 | = | -3 – 4 | = 7.

x ^ 2 + 5x + 4 = 0;

x12 = (-5 + – √25 – 4 * 4) / 2;

x1 = (-5 – 3) / 2 = -4; x2 = (-5 + 3) / 2 = 1;

| x1 + x2 | = | -4 + 1 | = 3.

Answer: the modulus of the sum of the roots of equations 7 and 3.