Set up the expression in this form. You cannot use synthetic division in this case.
So we need a value that multiplies by x^3 to get x^6. This value is x^3
__x^3_________________________
x^3+x^2-4x+3 | x^6 + 4x^5 + 0x^4 + 0x^3 - 3x^2 + 5x
-(x^6 + x^5 - 4x^4 + 3x^3 ) Product of x^3(x^3+x^2-4x+3)
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3x^5 + 4x^4 - 3x^3 - 3x^2 + 5x
Repeat for the 2nd part of the quotient. We need a value that multiplies by x^3 to get 3x^5, which is 3x^2
__x^3_+__3x^2_____________________
x^3+x^2-4x+3 | x^6 + 4x^5 + 0x^4 + 0x^3 - 3x^2 + 5x
-(x^6 + x^5 - 4x^4 + 3x^3 )
----------------
3x^5 + 4x^4 - 3x^3 - 3x^2
-(3x^5 + 3x^4 - 12x^3 + 9x^2) Product of 3x^2(x^3+x^2-4x+3)
------------------------------
x^4 + 9x^3 -12x^2 +5x
Repeat these steps until you get down to a irreducible remainder. Then add the remainder over the quotient to get your answer.
__x^3_+__3x^2_+__x___+__8__________________
x^3+x^2-4x+3 | x^6 + 4x^5 + 0x^4 + 0x^3 - 3x^2 + 5x + 0x^0
-(x^6 + x^5 - 4x^4 + 3x^3 )
----------------
3x^5 + 4x^4 - 3x^3 - 3x^2
-(3x^5 + 3x^4 - 12x^3 + 9x^2)
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x^4 + 9x^3 - 12x^2 +5x
-(x^4 + x^3 - 4x^2 +3x)
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8x^3 - 8x^2 +2x +0x^0
-(8x^3 + 8x^2 -32x +24 )
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-16x^2 + 32x - 24
So your answer should be