Let's perform the synthetic division
#color(white)(aaaa)##2##|##color(white)(aaaa)##-4##color(white)(aaaaa)##0##color(white)(aaaaaa)##3##color(white)(aaaaaaa)##-5#
#color(white)(aaaaa)##|##color(white)(aaaa)##color(white)(aaaaaa)##-8##color(white)(aaaa)##-16##color(white)(aaaaa)##-26#
#color(white)(aaaaaaaaa)###_________
#color(white)(aaaaa)##|##color(white)(aaaa)##-4##color(white)(aaaa)##-8##color(white)(aaaa)##-13##color(white)(aaaaa)##color(red)(-31)#
The remainder is #-31# and the quotient is #=-4x^2-8x-13#
#(-4x^3+3x-5)/(x-2)=-4x^2-8x-13-31/(x-2)#
Apply the remainder theorem
When a polynomial #f(x)# is divided by #(x-c)#, we get
#f(x)=(x-c)q(x)+r#
Let #x=c#
Then,
#f(c)=0+r#
Here,
#f(x)=-4x^3+3x-5#
Therefore,
#f(2)=-4*2^3+3*2-5#
#=-32+6-5#
#=-31#