b(a+2b) >ab-3

To solve the inequality \( b(a + 2b) > ab - 3 \), we can start by expanding the left-hand side:

1. Distributing \( b \) on the left side: \[ ba + 2b^2 > ab - 3 \]

2. Since \( ba \) is the same as \( ab \), we can rewrite the inequality as: \[ ab + 2b^2 > ab - 3 \]

3. Next, we can subtract \( ab \) from both sides: \[ 2b^2 > -3 \]

4. Now, to isolate \( b^2 \), divide both sides by 2: \[ b^2 > -\frac{3}{2} \]

Since \( b^2 \) is always non-negative (for real numbers), this inequality will always hold true for any real number \( b \). Thus, the solution set for \( b \) is all real numbers.

So the final statement is: \[ \text{All real numbers } b \text{ satisfy the inequality.} \]