Practice and apply :Factor each trinomial if it is not factorable .write prime

Let's practice factoring trinomials. I'll provide a few examples, and then I'll factor them or determine if they are prime (i.e., not factorable).

### Example 1: \( x^2 + 7x + 10 \)

To factor this trinomial, look for two numbers that multiply to 10 and add to 7. 

These numbers are 5 and 2. So, we can factor the trinomial as:

\[
x^2 + 7x + 10 = (x + 5)(x + 2)
\]

### Example 2: \( x^2 + 8x + 15 \)

Here, look for two numbers that multiply to 15 and add to 8.

These numbers are 5 and 3. So, the trinomial can be factored as:

\[
x^2 + 8x + 15 = (x + 5)(x + 3)
\]

### Example 3: \( x^2 + 6x + 9 \)

Look for two numbers that multiply to 9 and add to 6.

The numbers are 3 and 3. So, this trinomial is a perfect square:

\[
x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2
\]

### Example 4: \( x^2 + 5x + 8 \)

Look for two numbers that multiply to 8 and add to 5.

No two real numbers satisfy this condition, so this trinomial is **prime**.

### Example 5: \( 2x^2 + 7x + 3 \)

Here, look for two numbers that multiply to \( 2 \times 3 = 6 \) and add to 7.

These numbers are 6 and 1. However, because the coefficient of \( x^2 \) is 2, we factor by grouping:

\[
2x^2 + 7x + 3 = 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
\]

Would you like to try more examples or need help with a specific trinomial?

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Here are some related questions:
1. How do you factor trinomials with a leading coefficient greater than 1?
2. What is the difference between factoring by grouping and using the quadratic formula?
3. How do you determine if a trinomial is a perfect square?
4. What is the role of the discriminant in determining if a trinomial is factorable?
5. How can factoring be applied in solving quadratic equations?

**Tip:** When factoring trinomials, start by checking if the trinomial is a perfect square or can be factored easily before trying more complex methods like grouping or the quadratic formula.

Practice and apply the concept of factoring trinomials with detailed examples and solutions. Learn how to factor trinomials or determine if they are prime. Suitable for high school students learning algebraic concepts.

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