How to bend my solid in the right way?


Here is a way to do it, although it will probably get more deformed that you like to.
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Step 1:
First you will need to subdivide the part in order to keep the circular part straight. Draw a sketch like the pic and make a split line feature. Select all the faces of the part.
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Step 2:
You need to edit the 3d sketch and cut the spline so it is not close the region you want to remain unaffected.
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Step 3:
Create a deformed feature with curve to curve selected. Then select your initial curves on the first field and the 3d sketch in the second field.
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Step 4:
In the deform region tab check fixed edges and insert as fixed the surfaces of the round shape you want to remain straight.
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Step 5:
In shape options you will need to experiment to get the best result. Exit and see if your final shape is what you want. Good luck.