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 factorize $$7x^2 + 10x + 6$$. 4. **Use the factoring method:** For a quadratic $$ax^2 + bx + c$$, find two numbers that multiply to $$a \times c = 7 \times 6 = 42$$ and add to $$b = 10$$. The numbers are 6 and 7 because $$6 \times 7 = 42$$ and $$6 + 7 = 13$$, which is not 10, so try other pairs. Try 3 and 14: $$3 \times 14 = 42$$ and $$3 + 14 = 17$$ no. Try 2 and 21: $$2 \times 21 = 42$$ and $$2 + 21 = 23$$ no. Try 1 and 42: $$1 \times 42 = 42$$ and $$1 + 42 = 43$$ no. Try 5 and 8: $$5 \times 8 = 40$$ no. Try 7 and 6: already tried. Since no integer pairs add to 10, the quadratic does not factor nicely with integers. 5. **Use the quadratic formula to factor:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-10 \pm \sqrt{10^2 - 4 \times 7 \times 6}}{2 \times 7} = \frac{-10 \pm \sqrt{100 - 168}}{14} = \frac{-10 \pm \sqrt{-68}}{14}$$ 6. **Since the discriminant is negative, the quadratic has no real roots and cannot be factored over the reals.** **Final answer:** The expression simplifies to $$7x^2 + 10x + 6$$ and cannot be factorized further over the real numbers.