Introduction: In this problem, we are given a point (-1,2) through which a straight line passes and we need to find the equation of the line. We are also given that the distance of the line from the origin is 1 unit. To find the equation, we can use the distance formula and the equation of a straight line.

Approach:
1. Use the distance formula to find the distance between the origin (0,0) and the given point (-1,2).
2. Set the distance equal to 1 and solve for the unknown variable.
3. Use the equation of a straight line in point-slope form to find the equation of the line passing through the given point (-1,2) and with the slope found in the previous step.

Detailed Steps:
1. Use the distance formula to find the distance between the origin (0,0) and the given point (-1,2):
- The distance formula is given by: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Substitute the values: d = sqrt((0 - (-1))^2 + (0 - 2)^2)
- Simplify: d = sqrt(1 + 4) = sqrt(5)
- Set the distance equal to 1: sqrt(5) = 1
- Square both sides: 5 = 1^2
- Therefore, the distance between the origin and the given point is 1 unit.

2. Use the equation of a straight line in point-slope form to find the equation of the line passing through the given point (-1,2) and with the slope found in the previous step:
- The point-slope form of a straight line is given by: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
- Substitute the values: y - 2 = m(x - (-1))
- Simplify: y - 2 = mx + m
- Rearrange the equation: y = mx + m + 2
- Therefore, the equation of the line passing through the point (-1,2) is y = mx + m + 2.

Conclusion: The equation of the straight line passing through the point (-1,2) and with a distance of 1 unit from the origin is y = mx + m + 2.