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I have a set of polygons of different shapes and sizes in shapefile1.
What I need to do, is to create new polygons around the boundaries of the polygons in shapefile1, larger by x meters.
Try Buffer geoprocessing.
You can find it under Geoprocessing\buffer, on buffer window specify the shapefile of exciting polygons and the distance(scale factor) you want to buffer around existing polygons.
Use the buffer tool under geoprocessing menu. Input the first shapefile and state the distance you want. See the image attached.
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