Final answer:
The quadratic expression x' + 4x - 8 cannot be rewritten in the given form (x + p)^2 + 9 with correct values of p and q from the provided options, as completing the square results in different values than those listed.
Step-by-step explanation:
The question asks us to rewrite the quadratic expression x' + 4x - 8 in the form (x + p)^2 + q and find the values of p and q. First, we need to complete the square for the quadratic part of the expression.
We have:
- x^2 + 4x can be thought of as the beginning of an expanded (x + p)^2, where p is the number that when doubled gives us the middle term's coefficient, which is 4. So p = 2 because 2*2 = 4.
- When we expand (x + 2)^2, we get x^2 + 4x + 4. To get the original expression, we would need to subtract 12 from this to obtain -8. Therefore, q = -12.
However, none of the solution options match q = -12. There may have been an error in the original question, or we might need to adjust the given options. Since the constant part (-8) and the form given ((x + p)^2 + 9) don't perfectly correspond to a completion of a square followed by the addition of an arbitrary number (+9), the solutions provided in the options don't tie up with the expression x' + 4x - 8. The question as posed does not have a valid answer based on the standard method of completing the square. Thus, it's possible that there's a typo in the given expression or in the possible answers.