To factor the quadratic expression \( 2x^2 + 7x + 3 \), we need to find two numbers that multiply to the product of the coefficient of \( x^2 \) (which is 2) and the constant term (which is 3). This product is:
\[ 2 \times 3 = 6 \]
We also need these two numbers to add up to the coefficient of \( x \) (which is 7). The numbers that satisfy these conditions are 6 and 1, since:
\[ 6 \times 1 = 6 \quad \text{and} \quad 6 + 1 = 7 \]
Now, we can rewrite the middle term (7x) using these two numbers:
\[ 2x^2 + 6x + 1x + 3 \]
Next, we group the terms:
\[ (2x^2 + 6x) + (1x + 3) \]
Now, we factor out the common factors in each group:
\[ 2x(x + 3) + 1(x + 3) \]
Now, we can factor out the common binomial factor \((x + 3)\):
\[ (2x + 1)(x + 3) \]
Thus, the factored form of the quadratic expression \( 2x^2 + 7x + 3 \) is:
\[ \boxed{(2x + 1)(x + 3)} \]