1. The problem is to factorize the quadratic expression $$x^2 + 5x + 6$$. 2. Start by breaking the middle term into two terms whose coefficients add up to 5 and multiply to 6. Here, $$5x$$ is split into $$2x + 3x$$. 3. Rewrite the expression as $$x^2 + 2x + 3x + 6$$. 4. Group terms to factor by grouping: $$(x^2 + 2x) + (3x + 6)$$. 5. Factor out the greatest common factor from each group: $$x(x + 2) + 3(x + 2)$$. 6. Notice that $$x + 2$$ is a common binomial factor; factor it out: $$(x + 2)(x + 3)$$. Therefore, the factorized form of $$x^2 + 5x + 6$$ is $$ (x + 2)(x + 3) $$.