Divide using long division. State the quotient, \( \mathrm{q}(\mathrm{x}) \), and the remainder, \( \mathrm{r}(\mathrm{x}) \). \( \left(10 \mathrm{x}^{2}+3 \mathrm{x}-3\right) \div(2 \mathrm{x}-1) \) \( \left(10 \mathrm{x}^{2}+3 \mathrm{x}-3\right)+(2 \mathrm{x}-1)=\square+\frac{\square}{2 \mathrm{x}-1} \) \( ( \) Simplify your answers. Do not factor.)

To divide \( 10\mathrm{x}^2 + 3\mathrm{x} - 3 \) by \( 2\mathrm{x} - 1 \) using long division, follow these steps: 1. Divide the leading term of the dividend \( 10\mathrm{x}^2 \) by the leading term of the divisor \( 2\mathrm{x} \) to get \( 5\mathrm{x} \). This is the first term of the quotient \( \mathrm{q}(\mathrm{x}) \). 2. Multiply \( 5\mathrm{x} \) by \( 2\mathrm{x} - 1 \): \( 5\mathrm{x}(2\mathrm{x} - 1) = 10\mathrm{x}^2 - 5\mathrm{x} \). 3. Subtract this from the original polynomial: \( (10\mathrm{x}^2 + 3\mathrm{x} - 3) - (10\mathrm{x}^2 - 5\mathrm{x}) = 8\mathrm{x} - 3 \). 4. Now, repeat the process. Divide the leading term \( 8\mathrm{x} \) by \( 2\mathrm{x} \) to get \( 4 \) (the next term of the quotient). 5. Multiply \( 4 \) by \( 2\mathrm{x} - 1 \): \( 4(2\mathrm{x} - 1) = 8\mathrm{x} - 4 \). 6. Subtract these: \( (8\mathrm{x} - 3) - (8\mathrm{x} - 4) = 1 \). Now, we have the entire long division completed. The quotient \( \mathrm{q}(\mathrm{x}) = 5\mathrm{x} + 4 \) and the remainder \( \mathrm{r}(\mathrm{x}) = 1 \). Finally, we can express the result as: \[ (10 \mathrm{x}^{2}+3 \mathrm{x}-3) = (2 \mathrm{x}-1)(5\mathrm{x} + 4) + 1 \] So, the answer is: \[ \left(10 \mathrm{x}^{2}+3 \mathrm{x}-3\right)+(2 \mathrm{x}-1)=5\mathrm{x}+4+\frac{1}{2\mathrm{x}-1} \]