
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
1. A quadratic polynomial \( f(x) = x^2 + px + q \), with \( p \) and \( q \) real numbers, is said to be a double-up polynomial if it has two real roots, one of which is twice the other.
(a) If a double-up polynomial \( f(x) \) has \( p = -15 \), determine the value of \( q \).
(b) If \( f(x) \) is a double-up polynomial with one of the roots equal to 4, determine all possible values of \( p + q \).
(c) Determine all double-up polynomials for which \( p + q = 9 \).

Present the answer in LaTex format: \boxed{Your answer}
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