1. **State the problem:** Factor the polynomial $$x^4 + 4x^3 + 6x^2 + 8x + 8$$. 2. **Recall the formula:** The polynomial resembles the expansion of $(x+2)^4 = x^4 + 4x^3 + 6x^2 + 4x + 16$, but the last two terms differ. 3. **Try to factor by grouping or substitution:** Let's attempt to write the polynomial as $(x^2 + ax + b)^2 + c(x + d)$ or find a factorization. 4. **Check for possible factorization:** Try $(x^2 + 2x + 2)^2 = x^4 + 4x^3 + 8x^2 + 8x + 4$ which is close but not exact. 5. **Subtract to find difference:** $$x^4 + 4x^3 + 6x^2 + 8x + 8 - (x^2 + 2x + 2)^2 = (x^4 + 4x^3 + 6x^2 + 8x + 8) - (x^4 + 4x^3 + 8x^2 + 8x + 4) = -2x^2 + 4$$ 6. **Rewrite polynomial:** $$x^4 + 4x^3 + 6x^2 + 8x + 8 = (x^2 + 2x + 2)^2 - 2(x^2 - 2)$$ 7. **Use difference of squares:** $$= [(x^2 + 2x + 2) - \sqrt{2}(x - \sqrt{2})][(x^2 + 2x + 2) + \sqrt{2}(x - \sqrt{2})]$$ 8. **Simplify each factor:** - First factor: $$x^2 + 2x + 2 - \sqrt{2}x + 2 = x^2 + (2 - \sqrt{2})x + 4$$ - Second factor: $$x^2 + 2x + 2 + \sqrt{2}x - 2 = x^2 + (2 + \sqrt{2})x$$ 9. **Final factorization:** $$x^4 + 4x^3 + 6x^2 + 8x + 8 = (x^2 + (2 - \sqrt{2})x + 4)(x^2 + (2 + \sqrt{2})x)$$ --- **Maple T.A code for factorization:** `factor(x^4 + 4*x^3 + 6*x^2 + 8*x + 8);`