\( \frac{12 x^{4}+27 x^{3}-2 x^{2}}{3 x^{2}+1} \) \( \frac{12 x^{4}+27 x^{3}-2 x^{2}}{3 x^{2}+1}=\square+\frac{\square}{3 x^{2}+1} \) (Simplify your answers. Do not factor. Use integers or fractions for any numbers in the expressions.) \( l \)

To simplify \( \frac{12 x^{4}+27 x^{3}-2 x^{2}}{3 x^{2}+1} \), we can use polynomial long division. We divide the numerator by the denominator: 1. Divide the leading term of the numerator \( 12x^4 \) by the leading term of the denominator \( 3x^2 \) to get \( 4x^2 \). 2. Multiply \( 4x^2 \) by the entire denominator \( 3x^2 + 1 \): \( 4x^2(3x^2 + 1) = 12x^4 + 4x^2 \). 3. Subtract this result from the original numerator: \[(12x^4 + 27x^3 - 2x^2) - (12x^4 + 4x^2) = 27x^3 - 6x^2.\] 4. Now, repeat the process. Divide \( 27x^3 \) by \( 3x^2 \) to get \( 9x \). 5. Multiply \( 9x \) by \( 3x^2 + 1 \): \( 9x(3x^2 + 1) = 27x^3 + 9x \). 6. Subtract again: \[(27x^3 - 6x^2) - (27x^3 + 9x) = -6x^2 - 9x.\] 7. Finally, divide \( -6x^2 \) by \( 3x^2 \) to get \( -2 \). 8. Multiply: \[-2(3x^2 + 1) = -6x^2 - 2.\] 9. Subtract once more: \[(-6x^2 - 9x) - (-6x^2 - 2) = -9x + 2.\] In conclusion, we have: \[ \frac{12 x^{4} + 27 x^{3} - 2 x^{2}}{3 x^{2} + 1} = 4x^{2} + 9x - 2 + \frac{-9x + 2}{3x^{2} + 1} \] Thus, the final answer, filling in the squares: \[ \frac{12 x^{4} + 27 x^{3} - 2 x^{2}}{3 x^{2} + 1} = 4x^{2} + 9x - 2 + \frac{-9x + 2}{3 x^{2} + 1} \]