To factor the expression 2x² + 2x + 364, we can start by looking for a common factor in all of the terms.
1. **Identify the common factor:** In this case, we can see that each term is divisible by 2. So, we can factor out 2 from the expression:
2(x² + x + 182)
2. **Focus on the quadratic expression:** Now, we need to factor the quadratic expression x² + x + 182. To do this, we will look for two numbers that multiply to 182 and add to 1.
3. **Finding factors:** The possible pairs of factors of 182 include:
- 1 and 182
- 2 and 91
- 7 and 26
- 13 and 14
However, none of these pairs add up to 1. Thus, the quadratic expression x² + x + 182 does not factor neatly using rational numbers.
4. **Conclusion:** The simplest factorization we can achieve for the original expression is:
2(x² + x + 182). Since further factoring of x² + x + 182 is not possible with real numbers, we conclude that this is the fully factored form.
5. **Alternative methods:** If you’re looking for a solution that involves complex numbers or conic sections, you could use the quadratic formula. However, for most purposes, the expression in its current factored state suffices.