To factor the trinomial \( 3x^2 + 7x + 4 \), we can look for two numbers that multiply to \( 3 \cdot 4 = 12 \) (the product of the leading coefficient and the constant term) and add up to \( 7 \) (the middle coefficient).
The numbers that meet this criteria are \( 3 \) and \( 4 \) because:
- \( 3 \times 4 = 12 \)
- \( 3 + 4 = 7 \)
Next, we can rewrite the middle term \( 7x \) using these two numbers:
\[ 3x^2 + 3x + 4x + 4 \]
Now, we can group the terms:
\[ (3x^2 + 3x) + (4x + 4) \]
Now factor out the greatest common factors from each group:
\[ 3x(x + 1) + 4(x + 1) \]
Now we can factor out the common binomial factor \( (x + 1) \):
\[ (3x + 4)(x + 1) \]
Thus, the factored form of the trinomial \( 3x^2 + 7x + 4 \) is:
\[ (3x + 4)(x + 1) \]
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