To factor the quadratic expression $x^2 + 5x + 6$, we look for two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the linear term $x$).

The factors of 6 that add up to 5 are 2 and 3. Therefore, we can rewrite the expression as:

$x^2 + 5x + 6 = x^2 + 2x + 3x + 6$

Now, we group the terms:

$x^2 + 2x + 3x + 6 = (x^2 + 2x) + (3x + 6)$

Factor out the greatest common factor from each group:

$x(x + 2) + 3(x + 2)$

Notice that $(x + 2)$ is a common factor:

$(x + 2)(x + 3)$

So, the factored form of $x^2 + 5x + 6$ is:

$\boxed{(x + 2)(x + 3)}$

Would you like further details or have any questions about this process?

Related Questions

1. How do you factor a quadratic expression when the leading coefficient is not 1?
2. What are the methods to solve a quadratic equation?
3. How do you determine the roots of a quadratic equation from its factored form?
4. Can every quadratic expression be factored into real linear factors?
5. What is the relationship between the roots of a quadratic equation and its coefficients?

Tip: When factoring quadratics, always check if the quadratic is in standard form $(ax^2 + bx + c)$ and identify if there are simple factors before using more complex methods like completing the square or the quadratic formula.