Final answer:
To vertically stretch the function, multiply it by 2 and then translate it 4 units to the right.
Step-by-step explanation:
To vertically stretch the function by a factor of 2, we multiply the entire function by 2. The new function becomes: f(x) = 2(x² + 2x + 6).
To translate the function 4 units to the right, we subtract 4 from the x-values. The new function becomes: f(x) = 2((x - 4)² + 2(x - 4) + 6).
Therefore, the function f(x) = x² + 2x + 6, when vertically stretched by a factor of 2 and translated 4 units to the right, is represented by the function f(x) = 2((x - 4)² + 2(x - 4) + 6).