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

To divide \( x^{2}+11x+33 \) by \( x+7 \), we can use polynomial long division. 1. Divide the leading term of the dividend \( x^2 \) by the leading term of the divisor \( x \) to get \( x \). 2. Multiply the entire divisor \( x+7 \) by \( x \) to obtain \( x^2 + 7x \). 3. Subtract \( (x^2 + 7x) \) from \( (x^2 + 11x + 33) \): \[ (x^2 + 11x + 33) - (x^2 + 7x) = 4x + 33 \] 4. Now, divide the leading term \( 4x \) by the leading term of the divisor \( x \) to get \( 4 \). 5. Multiply the entire divisor \( x+7 \) by \( 4 \) to obtain \( 4x + 28 \). 6. Subtract \( (4x + 28) \) from \( (4x + 33) \): \[ (4x + 33) - (4x + 28) = 5 \] Now, we see that we can no longer divide because the degree of the remainder \( 5 \) is less than the degree of the divisor \( x+7 \). So, the quotient is \( x + 4 \) and the remainder is \( 5 \). Quotient: \( x + 4 \) Remainder: \( 5 \)