We are given the quadratic expression:

$2n^2 - 5n + 2$

We need to find an integer $n$ such that this expression is prime.

Step 1: Trial substitution for small integer values of $n$

We will substitute different integer values of $n$ into the expression and check if the result is a prime number.

For $n = 1$:

$2(1)^2 - 5(1) + 2 = 2 - 5 + 2 = -1 \quad (\text{not prime})$

For $n = 2$:

$2(2)^2 - 5(2) + 2 = 2(4) - 10 + 2 = 8 - 10 + 2 = 0 \quad (\text{not prime})$

For $n = 3$:

$2(3)^2 - 5(3) + 2 = 2(9) - 15 + 2 = 18 - 15 + 2 = 5 \quad (\text{prime!})$

So, when $n = 3$, the expression evaluates to 5, which is a prime number.

Thus, $n = 3$ is the integer that satisfies the condition.

Final Answer:

The integer $n$ is $3$.

Would you like more details on how to solve quadratic expressions? Here are five related questions:

1. Can other values of $n$ make the expression prime?
2. How do you factor quadratic equations in general?
3. What is the general process for identifying prime numbers?
4. How does the quadratic formula apply to this type of problem?
5. Can we create a similar expression where the solution yields larger primes?

Tip: Always check small integer values first when looking for specific solutions in quadratic expressions!