Find the quotient and the remainder (if any), when:

a³ - 5a² + 8a + 15 is divided by a + 1.

Dividing a³ - 5a² + 8a + 15 by a + 1

$\begin{array}{l} \phantom{a + 1)}{a^2 - 6a + 14} \ a + 1\overline{\smash{\big)}a^3 - 5a^2 + 8a + 15} \ \phantom{a + 1}\underline{\underset{-}{}a^3 \underset{-}{+}a^2} \ \phantom{{a + 1}2x}-6a^2 +8a + 15 \ \phantom{{a + 1}21x}\underline{\underset{+}{-}6a^2 \underset{+}{-} 6a} \ \phantom{{a + 1}{2x^3+}{+5x^2+}}14a + 15 \ \phantom{{a + 1}{2x^3+}{+5x^2}}\underline{\underset{+}{-}14a \underset{-}{+} 14} \ \phantom{{a + 1}{a^3 - 5a^2 + 8a + 152}} 1 \end{array}$

Quotient = a² - 6a + 14

Remainder = 1

Verification:

Quotient x Divisor + Remainder

= (a² - 6a + 14) $\times$ (a + 1) + 1

= a $\times$ (a² - 6a + 14) + 1 $\times$ (a² - 6a + 14) + 1

= a⁽¹⁺²⁾ - 6a⁽¹⁺¹⁾ + 14a + a² - 6a + 14 + 1

= a³ - 6a² + 14a + a² - 6a + 14 + 1

= a³ + (- 6a² + a²) + (14a - 6a) + (14 + 1)

= a³ - 5a² + 8a + 15

= Dividend