Let's simplify the expression step by step:

Distribute the terms within the parentheses:
a + (a+3)(a-3) - (a+2)(a²-2a+4)

Expand the expressions within the parentheses:
a + (a² - 3a + 3a - 9) - (a³ - 2a² + 4a + 2a² - 4a + 8)

Combine like terms:
a + a² - 3a + 3a - 9 - a³ + 2a² - 4a - 2a² + 4a - 8

Simplify:
a + a² - 3a + 3a - 9 - a³ + 2a² - 4a - 2a² + 4a - 8
a + a² - 3a + 3a - 9 - a³ + 2a² - 4a - 2a² + 4a - 8
a + a² - a³ - 9 - 8
a + a² - a³ - 17

Therefore, the simplified expression is a + a² - a³ - 17.