To factor the expression \( 2x^2 + 5x + 14x + 35 \), we can first combine the like terms:
\[ 2x^2 + (5x + 14x) + 35 = 2x^2 + 19x + 35 \]
Next, we need to factor the quadratic expression \( 2x^2 + 19x + 35 \). We are looking for two numbers that multiply to \( 2 \times 35 = 70 \) and add to \( 19 \).
The two numbers that fit these criteria are \( 14 \) and \( 5 \).
Now, we can rewrite the expression:
\[ 2x^2 + 14x + 5x + 35 \]
Next, we can group the terms:
\[ (2x^2 + 14x) + (5x + 35) \]
Now we factor out the common factors from each group:
\[ 2x(x + 7) + 5(x + 7) \]
Now we notice that \( (x + 7) \) is a common factor:
\[ (2x + 5)(x + 7) \]
So, the factored form of the expression \( 2x^2 + 19x + 35 \) is:
\[ \boxed{(2x + 5)(x + 7)} \]
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