Final answer:
George H.W. Bush faced the challenge of reducing the federal budget deficit while attempting to keep his promise of not raising taxes.
Step-by-step explanation:
The challenge George H.W. Bush experienced when he first took office was how to address the federal budget deficit that had significantly increased during the Reagan administration. Despite his promise of 'Read my lips: no new taxes,' President Bush found himself in the difficult position of needing to reduce the deficit without breaking his tax pledge. This led to a bipartisan agreement that included both spending cuts and mild tax increases, which subsequently cost him political credibility among his Republican colleagues.