The answer is (x + 1)^2 + 2.
This is a concept I always found very confusing when first learning quadratics, so don’t worry about the struggle.
To explain the reasoning, we should look at the clues we’re given by the table. Since we know we’re looking at a quadratic function, we know that we’re going to have a point that either serves as a min or a max, where the values will go from increasing to decreasing or vise versa. I’m this example, we see the values go from 6, to 3, to 2 (decreasing) and then to 3, and 6 and so on (increasing). This means our minimum is located at h(x) = 2, x = -1 for this function.
Because our minimum is on at x = 0, which is where it would be normally, we can assume there is a value affecting the expression being squared. To find the value, we need to set x to our current minimum, which is -1(x = -1). In order to figure out how our value is being affected, we want to set the left side with our x to equal zero. To do this we would add 1 to both sides (x + 1) = (-1 + 1) which would give us (x + 1) = 0. We now know the value that is being squared in the function, (x + 1)^2.
To find the value added outside of the squared expression, we simply look at h(x) at our minimum location. In this case h(-1) = 2. Normally a quadratic function will have its minimum value be 0, so the outside of the squared expression is being added by 2, giving us (x + 1)^2 + 2.
I’m sorry if this isn’t explained very well. I was trying to write it in a way I thought would be easier to understand, but I’m not sure if it will come across that way. If you need further explanation, please let me know.