Answer:
[tex]\displaystyle \boxed{x^2 - x + 1}[/tex]
Step-by-step explanation:
Divide the leading coefficients of the divisor and dividend:
[tex]\displaystyle x^2 = \frac{x^4}{x^2} \\ \\ x^4 + x^3 - 3x^2 = x^2[x^2 + x - 3][/tex]
[tex]\displaystyle -x^3 + 4x - 3 = -[x^4 + x^3 - 3x^2] + [x^4 - 3x^2 + 4x - 3][/tex]
Now you have this:
[tex]\displaystyle \frac{-x^3 + 4x - 3}{x^2 + x - 3} \\ \\ -x = \ -\frac{x^3}{x^2} \\ \\ 1 = \frac{-3}{-3}[/tex]
Altogether, you have this:
[tex]\displaystyle \boxed{x^2 - x + 1}[/tex]
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