Divide. \[ \left(8-55 x+42 x^{2}\right) \div(6 x-1) \]

To divide the polynomial \( 8 - 55x + 42x^2 \) by \( 6x - 1 \), we can use polynomial long division. 1. **Setup the division**: Write \( 42x^2 - 55x + 8 \) under the long division symbol and \( 6x - 1 \) outside. 2. **First division**: Divide the leading term \( 42x^2 \) by \( 6x \) to get \( 7x \). This is the first term of the quotient. 3. **Multiply**: Multiply \( 7x \) by \( 6x - 1 \): \[ 7x(6x - 1) = 42x^2 - 7x \] 4. **Subtract**: Subtract this from \( 42x^2 - 55x + 8 \): \[ (42x^2 - 55x + 8) - (42x^2 - 7x) = -55x + 7x + 8 = -48x + 8 \] 5. **Second division**: Now, divide the leading term \(-48x\) by \(6x\): \[ -48x \div 6x = -8 \] So, the next term of the quotient is \(-8\). 6. **Multiply**: Multiply \(-8\) by \(6x - 1\): \[ -8(6x - 1) = -48x + 8 \] 7. **Subtract**: Again, subtract this from \(-48x + 8\): \[ (-48x + 8) - (-48x + 8) = 0 \] Since the remainder is \(0\), the division ends here. The result of the division is: \[ \frac{8 - 55x + 42x^2}{6x - 1} = 7x - 8 \] Thus, the final answer is: \[ 7x - 8 \]