To find the full solution set for the polynomial equation \( x^3 + 5x^2 + 10x + 50 = 0 \) given that one solution is \( x = -5 \), we can first use polynomial long division or synthetic division to factor the polynomial by \( x + 5 \).
Step 1: Synthetic division
Perform synthetic division with \( x = -5 \):
-5 | 1 5 10 50
| -5 0 -50
----------------------
1 0 10 0
This gives us the quotient \( x^2 + 10 \) (the remainder is \( 0 \)), meaning we can factor the original cubic equation as:
\[ x^3 + 5x^2 + 10x + 50 = (x + 5)(x^2 + 10) \]
Step 2: Solve the quadratic factor
Now, we solve the quadratic equation \( x^2 + 10 = 0 \):
\[ x^2 = -10 \]
Taking the square root of both sides, we get:
\[ x = \pm i\sqrt{10} \]
Step 3: Summary of solutions
Thus, the full solution set for the equation \( x^3 + 5x^2 + 10x + 50 = 0 \) is:
\[ x = -5, \quad x = i\sqrt{10}, \quad x = -i\sqrt{10} \]
Step 4: Identify the nature of the solutions
- One real solution: \( -5 \)
- Two non-real solutions: \( i\sqrt{10} \) and \( -i\sqrt{10} \)
Conclusion:
The correct option that describes the solution set is:
Option #1: It has one real solution and two non-real solutions.