Final answer:
For the system of equations to have exactly one solution, the value of k must be where the horizontal line touches the vertex of the parabola. This occurs at k=-1, as the parabola y=(x+2)^2-1 has its vertex at (-2, -1).
Step-by-step explanation:
To determine for what value(s) of k the system of equations y=(x+2)^2-1 and y=k has exactly one solution, we need to understand that the first equation is a parabola that opens upwards, and the second equation is a horizontal line.
The parabola has a vertex at (-2, -1). The coordinate of the vertex suggests that if k is less than -1, the horizontal line y=k would not intersect the parabola, meaning there would be no solutions. If k is exactly equal to -1, the horizontal line would touch the vertex of the parabola, resulting in exactly one solution. For k values greater than -1, the horizontal line would intersect the parabola at two points, resulting in two solutions.
Therefore, the value of k for which there is exactly one solution is k=-1.