When the rate goes down, the price of Eurodollar actually goes up. So, if after aggregating, we have to pay floating rate and receive the fixed, then we actually needs to sell Eurodollar futures.
In other words, when the floating rate goes up, on one hand, we pay more as we have the obligation to pay floating; on the other hand, we receive the profit from shorting the Eurodollar, since the price of Eurodollar tends to go down. Thus, we can hedge the position.
Back to our problem, the total exposure, measured using duration should be 
DV01 of the portfolio=($420,000,000*4.433/1.0415-$385,000,000*7.581/1.0538)*0.01%=-98,200.48
Since DV01 of Eurodollar futures is $25, we should trade 98,200/25=3928 contracts
And the direction: remember that we have more floating payments than fixed, for the 10 years swap is receiving fixed while paying floating. So we need to short the Eurodollar contracts.
The closest answer would be B.  
Am I right?^^