Let a firm's cost function be given by c=20+4q+4q^2

Finding the optimal level of output

The marginal revenue is $9. The marginal cost is the derivative of the total cost with respect to q. Hence, the marginal cost is MC=4+8q. At the optimal level of output, the marginal revenue equals the marginal cost, i.e., 9=4+8q. Therefore, the optimal level of output is q=0.625.

Finding the firm's producer surplus

The producer surplus can be found as (Revenue at q=0.625) - (Variable cost at q=0.625). The variable cost is the part of the total cost which depends on q, i.e., it is 4q+4q^2. Hence, we obtain (9*0.625)-(4*0.625+4*0.625^2) = $1.56. The firm's producer surplus is $1.56.

Finding the firm's profit in the short run

To find its profit, we need to subtract the fixed cost from the producer surplus, which gives us $1.56 - $20 = -$18.44. Hence, the firm will generate a loss of $18.44 in the short run.

Please note that in the long run, the firm will leave the industry as it does not cover its fixed cost.