[tex]\frac{b^4-9b^2+3\sqrt[2]{3}b-1}{3b^2-2\sqrt[2]{3}b+1}[/tex]

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To simplify this expression, we can factor the numerator first:

[tex]b^4-9b^2+3\sqrt{3}b-1 = (b^2 - \sqrt{3})(b^2 + \sqrt{3}) - 9b^2 + 3\sqrt{3}b - 1[/tex]
[tex]= (b^2 - \sqrt{3})(b^2 + \sqrt{3}) - (3b - 1)^2[/tex]
[tex]= (b^2 - \sqrt{3})(b^2 + \sqrt{3} - (3b - 1))(3b - 1)[/tex]

Now the expression becomes:

[tex]\frac{(b^2 - \sqrt{3})(b^2 + \sqrt{3} - (3b - 1))(3b - 1)}{3b^2-2\sqrt{3}b+1}[/tex]

At this point, you can simplify further if needed, but this is the factored form of the original expression.