1. **State the problem:** Simplify the expression $$\frac{x^5 - 32}{x + 2}$$. 2. **Recall the formula:** This is a division of a polynomial by a binomial. Since the numerator is a difference of powers, we can use the factorization for difference of powers: $$a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$$. 3. **Identify terms:** Here, $$a = x$$ and $$b = 2$$, so: $$x^5 - 2^5 = (x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)$$. 4. **Rewrite numerator:** $$x^5 - 32 = (x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)$$. 5. **Divide by denominator:** $$\frac{x^5 - 32}{x + 2} = \frac{(x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)}{x + 2}$$. 6. **Note:** The denominator is $$x + 2$$, not $$x - 2$$, so the factor $$x - 2$$ in numerator does not cancel with denominator. 7. **Use polynomial division:** Divide $$x^5 - 32$$ by $$x + 2$$ using synthetic or long division. 8. **Long division steps:** - Divide $$x^5$$ by $$x$$ to get $$x^4$$. - Multiply $$x^4(x + 2) = x^5 + 2x^4$$. - Subtract: $$(x^5 - 32) - (x^5 + 2x^4) = -2x^4 - 32$$. - Divide $$-2x^4$$ by $$x$$ to get $$-2x^3$$. - Multiply $$-2x^3(x + 2) = -2x^4 - 4x^3$$. - Subtract: $$(-2x^4 - 32) - (-2x^4 - 4x^3) = 4x^3 - 32$$. - Divide $$4x^3$$ by $$x$$ to get $$4x^2$$. - Multiply $$4x^2(x + 2) = 4x^3 + 8x^2$$. - Subtract: $$(4x^3 - 32) - (4x^3 + 8x^2) = -8x^2 - 32$$. - Divide $$-8x^2$$ by $$x$$ to get $$-8x$$. - Multiply $$-8x(x + 2) = -8x^2 - 16x$$. - Subtract: $$(-8x^2 - 32) - (-8x^2 - 16x) = 16x - 32$$. - Divide $$16x$$ by $$x$$ to get $$16$$. - Multiply $$16(x + 2) = 16x + 32$$. - Subtract: $$(16x - 32) - (16x + 32) = -64$$. 9. **Result:** Quotient is $$x^4 - 2x^3 + 4x^2 - 8x + 16$$ and remainder is $$-64$$. 10. **Express division:** $$\frac{x^5 - 32}{x + 2} = x^4 - 2x^3 + 4x^2 - 8x + 16 - \frac{64}{x + 2}$$. **Final answer:** $$\boxed{\frac{x^5 - 32}{x + 2} = x^4 - 2x^3 + 4x^2 - 8x + 16 - \frac{64}{x + 2}}$$