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The question asks whether Ali is correct in thinking that the value of \( E = n^2 + n + 5 \) will always be a prime number for any whole number \( n \).

To check this, let's examine some values of \( n \) and see whether the resulting values of \( E \) are prime numbers:

1. For \( n = 0 \):
   \[
   E = 0^2 + 0 + 5 = 5
   \]
   \( 5 \) is a prime number.

2. For \( n = 1 \):
   \[
   E = 1^2 + 1 + 5 = 7
   \]
   \( 7 \) is a prime number.

3. For \( n = 2 \):
   \[
   E = 2^2 + 2 + 5 = 11
   \]
   \( 11 \) is a prime number.

4. For \( n = 3 \):
   \[
   E = 3^2 + 3 + 5 = 17
   \]
   \( 17 \) is a prime number.

5. For \( n = 4 \):
   \[
   E = 4^2 + 4 + 5 = 25
   \]
   \( 25 \) is **not** a prime number (since \( 25 = 5 \times 5 \)).

Therefore, Ali is **not correct**, as the expression does not always produce a prime number for any whole number \( n \) (in this case, \( n = 4 \) results in a non-prime value).

Would you like further details or have any questions?

Here are 5 related questions to expand on the topic:

1. What is the definition of a prime number?
2. How can you check if a large number is prime?
3. Are there other polynomial expressions that generate only prime numbers?
4. What is the significance of prime numbers in number theory?
5. How does the concept of prime numbers relate to cryptography?

**Tip:** To verify if a number is prime, check if it is divisible by any number other than 1 and itself up to its square root.

Ali thinks that the value of E will be a prime number for any whole number value of n. Is Ali correct? You must give a reason for your answer.

Explore whether the expression E = n^2 + n + 5 always results in a prime number for any whole number n. This problem delves into quadratic equations and prime numbers, offering a clear explanation with examples for middle and high school students.

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