1. **Stating the problem:** Divide the polynomial $2x^4 - 4x^3 - 5x^2 - 5$ by $x^2 + 2$. 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by this result, subtract, and repeat with the remainder. 3. **Step-by-step division:** - Divide the leading term $2x^4$ by $x^2$ to get $2x^2$. - Multiply the divisor by $2x^2$: $(x^2 + 2)(2x^2) = 2x^4 + 4x^2$. - Subtract this from the dividend: $$ (2x^4 - 4x^3 - 5x^2 - 5) - (2x^4 + 4x^2) = \cancel{2x^4} - 4x^3 - 5x^2 - 5 - \cancel{2x^4} - 4x^2 = -4x^3 - 9x^2 - 5 $$ - Bring down the remainder $-4x^3 - 9x^2 - 5$. - Divide the leading term $-4x^3$ by $x^2$ to get $-4x$. - Multiply the divisor by $-4x$: $(x^2 + 2)(-4x) = -4x^3 - 8x$. - Subtract this from the remainder: $$ (-4x^3 - 9x^2 - 5) - (-4x^3 - 8x) = \cancel{-4x^3} - 9x^2 - 5 - \cancel{-4x^3} + 8x = -9x^2 + 8x - 5 $$ - Bring down the new remainder $-9x^2 + 8x - 5$. - Divide the leading term $-9x^2$ by $x^2$ to get $-9$. - Multiply the divisor by $-9$: $(x^2 + 2)(-9) = -9x^2 - 18$. - Subtract this from the remainder: $$ (-9x^2 + 8x - 5) - (-9x^2 - 18) = \cancel{-9x^2} + 8x - 5 - \cancel{-9x^2} + 18 = 8x + 13 $$ 4. **Result:** The quotient is $2x^2 - 4x - 9$ and the remainder is $8x + 13$. 5. **Final answer:** $$\frac{2x^4 - 4x^3 - 5x^2 - 5}{x^2 + 2} = 2x^2 - 4x - 9 + \frac{8x + 13}{x^2 + 2}$$