Question

Without actual division p(x) = x4 + 2x3 - 2x2 + 2x - 3 is exactly divisible by g(x) = x2 + 2x - 3

Answer 1

now if f(x) is divisible by g(x) then g(x)=0
g(x)=x²+2x-3
0 =x²+3x-x-3
0 =x(x+3)-1(x+3)
0 =(x+3)(x-1)
either x+3=0
x=-3
or x-1=0
x=1
using x=-3
p(-3)= (-3)⁴+(2×-3)³-(2×-3)²+2(-3)-3
=81-54-18-6-3
=0
hence x=-3 is a solution for p(x)
using x=1
p(1)=(1)⁴+2(1)³-2(1)²+2(1)-3
=1+2-2+2-3
=0
hence x=1 is also a solution for p(x)
since for both the possible values for g(x) satisfies it to be a factor of p(x) . we can conclude that p(x) is exactly divisible by g(x)

Answer 2

Please see the attachment