Subtract two trinomials with like terms.

1. Distribute the negative sign: The expression involves subtracting the second trinomial from the first. This means we need to distribute the negative sign to each term within the second trinomial. So, $-( -3a^2 + 2b^2 + 8ab )$ becomes $3a^2 - 2b^2 - 8ab$. The problem now is to add the first trinomial to this result: $(3a^2 - 5ab + b^2) + (3a^2 - 2b^2 - 8ab)$.
2. Group like terms: Identify terms with the same variables raised to the same powers. In this expression, the like terms are $3a^2$ and $3a^2$, $-5ab$ and $-8ab$, and $b^2$ and $-2b^2$.
3. Combine like terms: Add the coefficients of the like terms.
- For the $a^2$ terms: $3a^2 + 3a^2 = 6a^2$
- For the $ab$ terms: $-5ab - 8ab = -13ab$
- For the $b^2$ terms: $b^2 - 2b^2 = -b^2$
4. Write the simplified expression: Combine the results from step 3 to form the final simplified trinomial.