Divide. If the polynomial does not divide evenly, include the remainder as a fraction. \[ \left(-17 t^{2}+166 t+31\right) \div(t-10) \] \begin{tabular}{|lll}\hline\( \square \) & \( \square \) \\ Submit \end{tabular}

First, let's perform polynomial long division. Divide the leading term of \(-17t^2\) by the leading term of \(t - 10\) to get \(-17t\). Multiply \(-17t\) by \((t - 10)\) to get \(-17t^2 + 170t\). Now, subtract this from the original polynomial: \[ (-17t^2 + 166t + 31) - (-17t^2 + 170t) = (166t - 170t) + 31 = -4t + 31. \] Next, divide the new leading term \(-4t\) by the leading term \(t - 10\) to get \(-4\). Multiply \(-4\) by \((t - 10)\) to obtain \(-4t + 40\). Now subtract: \[ (-4t + 31) - (-4t + 40) = 31 - 40 = -9. \] So, the final result of the division is \(-17t - 4\) with a remainder of \(-9\). Therefore: \[ \left(-17 t^{2}+166 t+31\right) \div(t-10) = -17t - 4 - \frac{9}{t-10}. \] That’s your answer! Putting it in the specified format: First additional information: Polynomial long division is like a math scavenger hunt! You find the biggest prize (the terms you can divide) first and keep track of what you've collected (the remainders). Just be careful not to lose any terms along the way! Second additional information: To check your work, you can easily multiply the quotient by the divisor and add the remainder. If you end up back at the original polynomial, you've hit the jackpot! If not, you might need to retrace your steps—just like in treasure hunting!