Question

1. $(a^3 + 8a^2 + 3a - 61) \div (a + 6)$

Explanation:

Step1: Set up synthetic division

Use root $a=-6$, coefficients $1, 8, 3, -61$

Step2: Bring down leading coefficient

$\downarrow 1$

Step3: Multiply and add to next coefficient

$1 \times (-6) = -6$; $8 + (-6) = 2$

Step4: Repeat multiply-add step

$2 \times (-6) = -12$; $3 + (-12) = -9$

Step5: Final multiply-add step

$-9 \times (-6) = 54$; $-61 + 54 = -7$

Step6: Write quotient and remainder

Quotient: $a^2 + 2a - 9$, Remainder: $-7$

Answer:

$a^2 + 2a - 9 - \frac{7}{a+6}$