Question 165600
P(x)=5x^4-3x^3+ax^2+bx-7

i is a zero of P(x) then -i is a zero of P(x) too
Using synthetic division:

Please see source code

   5    -3         a         b              -7
 i      5i      -5-3i       -5i+3+ai       5+3i-a+bi
   5    5i-3    -5-3i+a      -5i+3+ai+b    -2-a+(3+b)i=0
-i      -5i       3i           5i-ai
   5     -3      -5+a          3+b=0

-2-a+(3+b)i=0
3+b=0

then b=-3 and a=-2

P(x)=5x^4-3x^3-2x^2-3x-7