\( \left(x^{2}+11 x+33\right) \div(x+7) \) Your answer should give the quotient and the remainder. Quotient: \( \square \) Remainder: \( \square \)

We want to divide \[ x^2 + 11x + 33 \] by \[ x + 7. \] **Step 1.** Set up the division so that \[ \frac{x^2 + 11x + 33}{x + 7}. \] **Step 2.** Divide the leading term of the dividend, \(x^2\), by the leading term of the divisor, \(x\), to get the first term of the quotient: \[ \frac{x^2}{x} = x. \] **Step 3.** Multiply the divisor by \(x\) and subtract from the dividend: \[ x(x+7) = x^2 + 7x. \] Subtract from the original polynomial: \[ (x^2 + 11x + 33) - (x^2 + 7x) = 4x + 33. \] **Step 4.** Now, divide the new leading term \(4x\) by \(x\) to get the next term: \[ \frac{4x}{x} = 4. \] **Step 5.** Multiply the divisor by \(4\): \[ 4(x+7) = 4x + 28. \] Subtract: \[ (4x + 33) - (4x + 28) = 5. \] **Step 6.** Since \(5\) is of lower degree than the divisor \(x+7\), it is the remainder. Thus, the quotient is \[ x + 4, \] and the remainder is \[ 5. \] Quotient: \(x + 4\) Remainder: \(5\)