PowerShell Core, 103102 bytes
$s=-1
switch -r($args[$args.LengthCount..0]){.{$a=-$a;$s=-$s}~{$a--}}"$s`x+$a"|% *ce '+-','-'|% *ce '1x','x'
Passes an expression as a string using splatting
Then processes it from right to left
For each char, negate $s
and $a
, and if the char is ~
, decrement $a
Then concatenate them all: $s + "x+" + $a
Replace +-
with -
Remove the useless 1 before x