To simplify the expression (-12x^2/7 + 8/9 : 3 1/2 - 2/7 * 5/18) * 3 1/2, we need to follow the order of operations (PEMDAS/BODMAS).

First, let's simplify the expression inside the parentheses:
-12x^2/7 + 8/9 : 3 1/2 - 2/7 * 5/18

To simplify the division, we need to find a common denominator for 7 and 2. The least common multiple (LCM) of 7 and 2 is 14.
So, we can rewrite the expression as:
-12x^2/7 + 8/9 : 7/2 - 2/7 * 5/18

Now, let's simplify the division:
-12x^2/7 + 8/9 * 2/7 - 2/7 * 5/18

Multiply the numerators and denominators separately:
-24x^2/49 + 16/63 - 10/126

To add or subtract fractions, we need to find a common denominator. The LCM of 49, 63, and 126 is 882.
So, we can rewrite the expression as:
(-24x^2 * 18 + 16 * 14 - 10 * 7) / 882

Simplifying the numerator:
(-432x^2 + 224 - 70) / 882

Combining like terms:
(-432x^2 + 154) / 882

Now, let's multiply the simplified expression by 3 1/2:
(-432x^2 + 154) / 882 * 3 1/2

To multiply a fraction by a whole number, we multiply the numerator by the whole number and keep the denominator the same:
(-432x^2 + 154) * 7 / (882 * 2)

Simplifying the denominator:
(-432x^2 + 154) * 7 / 1764

Multiplying the numerator:
-3024x^2 + 1078 / 1764

So, the simplified expression is -3024x^2 + 1078 / 1764.