To factor the quadratic expression \(x^2 + 3x + 2\) as the product of two binomials, we look for two numbers that multiply to the constant term (which is \(2\)) and add to the coefficient of the linear term (which is \(3\)).
The two numbers that satisfy these conditions are \(1\) and \(2\):
- \(1 \times 2 = 2\)
- \(1 + 2 = 3\)
Now we can write the expression as the product of two binomials:
\[ x^2 + 3x + 2 = (x + 1)(x + 2) \]
Thus, the factored form is:
\[ \boxed{(x + 1)(x + 2)} \]
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