simplify

a-1/a
___________
2+3/2+1/a^2

To simplify the given expression, we can follow the order of operations (PEMDAS/BODMAS):

1. Start by simplifying the numerator, a - 1:
To add or subtract fractions, we need a common denominator. In this case, the denominator is "a". To make the denominator of "a-1" the same as "a", we can multiply the numerator and denominator by "a". This gives us (a * a - 1 * a) / a:

(a^2 - a) / a

2. Moving on to the denominator, 2 + 3/2 + 1/a^2:
Firstly, focus on the fraction, 3/2. Since the integer (2) is less than the denominator (2), we can rewrite the fraction as a mixed number by dividing 3 by 2:

3 ÷ 2 = 1 with a remainder of 1
So, 3/2 = 1 and 1/2

Now, we can rewrite the expression with the mixed number:

2 + 1 and 1/2 + 1/a^2

Now, let's simplify the addition:

2 + 1 = 3

Therefore, the expression becomes:

3 + 1/2 + 1/a^2

3. Finally, let's simplify the expression as a whole:
To add fractions, we need a common denominator. In this case, the least common denominator (LCD) is "2a^2". To make the denominators of the fractions equal to "2a^2", we can multiply the numerator and denominator of each fraction by the appropriate factors:

3 * (a^2 / a^2) + 1/2 * (a^2 / a^2) + 1/a^2

This gives us:

(3a^2) / (2a^2) + (1/2) * (a^2 / a^2) + 1/a^2

Simplifying further:

(3a^2) / (2a^2) + (a^2 / 2a^2) + 1/a^2

Now, we can combine the fractions:

(3a^2 + (a^2) + 1) / 2a^2

Simplifying the numerator:

(3a^2 + a^2 + 1) / 2a^2

(4a^2 + 1) / 2a^2

Therefore, the simplified expression is:

(4a^2 + 1) / 2a^2