To solve this inequality, we need to first simplify the expression:

ig(2x²+21x+9) - lg(2x+1) > 1

First, we can rewrite the expression using log properties:

log[((2x²+21x+9)/(2x+1))] > 1

Next, we can convert the inequality to its exponential form:

10^1 > (2x²+21x+9)/(2x+1)

10 > (2x²+21x+9)/(2x+1)

Now, we can multiply both sides by (2x+1) to get rid of the denominator:

10(2x+1) > 2x² + 21x + 9

Expand and simplify:

20x + 10 > 2x² + 21x + 9

Rearrange and set the inequality to zero:

0 > 2x² + x - 1

Now we need to solve this quadratic inequality. We can factor it or use the quadratic formula to find the critical points, then test intervals to determine the solution set.

Hope this helps! Let me know if you need further assistance.