Am I answering this correctly?

As a firm takes on more debt financing, risk of default and consequently bankruptcy increases. As the risk increases, a higher return on invetments is needed, which increases the overall cost of capital.

You've on the right track with your thinking, but unless I'm misreading, I think your chain of logic might be a bit off. It sounds like you're suggesting that the greater bankruptcy / financial distress risk--as created by the higher debt levels--drives management to seek out higher ROA, to which the capital providers, in turn, respond with higher COC demands. But note that an increasing COC would in turn push management into ever-riskier assets, and they quickly find themselves in a form of death spiral.

In theory, the opposite should be true: higher leverage demands more stability in earnings; hence it demands that a responsible management moves into higher-quality, lower-risk assets--which is directly correlated, of course, with a lower ROA.

So in deriving your answer to the question, let's leave out the ROA bit, and see if we can find a more direct cause-and-effect linkage between leverage and COC that doesn't involve ROA on corporate assets. First, you're correct that greater leverage (significantly beyond an optimal level) increases the risk of financial distress--and thus the probability of incurring the costs that distress / bankruptcy entails. (Cause)

These costs would be borne by some or all capital providers, depending on the situation (even reaching the most senior debt-holders, once the leverage is so great that the balance sheet is upside down). So for the next step in your answer, think what COC represents: the return demanded by claimholders to compensate them for the risks to which their investments are exposed. (Effect)