Determine all real values of the parameter 'a' for which the equation `16x^4 - ax^3 + (2a + 17) x^2 - ax + 16 = 0` has exactly four distinct real roots that form a geometric progression.

Answer

To determine all real values of the parameter 'a' for which the equation

\[ 16x^4 - ax^3 + (2a + 17)x^2 - ax + 16 = 0 \]

has exactly four distinct real roots that form a geometric progression, we can follow these steps:

### Step 1: Set Up the Roots
Assume the roots of the polynomial are in a geometric progression. Let the roots be \( b, bq, bq^2, bq^3 \), where \( b \) is the first term and \( q \) is the common ratio.