To factor the trinomial \( 2n^2 + 5n + 2 \), we need to break it down into two binomials.
We can look for two numbers that multiply to \( 2 \times 2 = 4 \) (the coefficient of \( n^2 \) multiplied by the constant term) and add up to \( 5 \) (the coefficient of \( n \)).
The numbers \( 4 \) and \( 1 \) work because:
- \( 4 \times 1 = 4 \)
- \( 4 + 1 = 5 \)
Now, we can rewrite the middle term \( 5n \) using \( 4n \) and \( 1n \): \[ 2n^2 + 4n + 1n + 2 \]
Next, we group the terms: \[ (2n^2 + 4n) + (1n + 2) \]
Now, we factor each group: \[ 2n(n + 2) + 1(n + 2) \]
We see that \( (n + 2) \) is a common factor: \[ (2n + 1)(n + 2) \]
So, the factored form of the trinomial \( 2n^2 + 5n + 2 \) is: \[ (2n + 1)(n + 2) \]
Among the provided choices, the correct one is: (2n + 1)(n + 2).