1. **State the problem:** Factorize the expression $$x^2 + 5x + 6x^2 + 5x + 6$$. 2. **Combine like terms:** Group the terms with the same powers of $x$. $$x^2 + 6x^2 + 5x + 5x + 6 = (x^2 + 6x^2) + (5x + 5x) + 6 = 7x^2 + 10x + 6$$ 3. **Factor the quadratic expression:** We want to factor $$7x^2 + 10x + 6$$. 4. **Use the factoring method for quadratics:** For $$ax^2 + bx + c$$, find two numbers that multiply to $$a \times c = 7 \times 6 = 42$$ and add to $$b = 10$$. 5. **Find the pair:** The numbers 6 and 7 multiply to 42 and add to 13, which is too high. The numbers 3 and 14 multiply to 42 and add to 17, also too high. The numbers 2 and 21 multiply to 42 and add to 23, too high. The numbers 1 and 42 multiply to 42 and add to 43, too high. The numbers 5 and 8 multiply to 40, not 42. The numbers 6 and 7 are closest but do not add to 10. 6. **Since no integer pair works, check for factorization by grouping or use quadratic formula:** 7. **Quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-10 \pm \sqrt{10^2 - 4 \times 7 \times 6}}{2 \times 7} = \frac{-10 \pm \sqrt{100 - 168}}{14} = \frac{-10 \pm \sqrt{-68}}{14}$$ 8. **Since the discriminant is negative, the quadratic does not factor over the real numbers.** **Final answer:** The expression simplifies to $$7x^2 + 10x + 6$$ and cannot be factorized further over the real numbers.