It’s a bit of a trick question that a number of skilled and experienced CEOs have had to answer in the past decade or so, and most of them came up with the wrong answers. But it’s a very real question for Meg Whitman, HP’s current CEO, and for everyone who works for her.

No one has come closer than Whitman to cracking that corporate koan. Yet recent evidence suggests she’s losing ground. Notably, HP is being dropped from the Dow Jones Industrial Average this month, since Visa (V) appears to be a better proxy for big tech. HP joined the Dow as part of the Class of 1997, along with Wal-Mart (WMT), Travelers (TRV), and Johnson & Johnson (JNJ). It is the first dropout of that class.

Part of HP’s Dow exit is because tech companies seem to age faster than retail, insurance, or health care companies. But more of it is because HP seems to have slid into an irreversible decline even before Whitman — or anyone — could have saved the company. If that’s the case, the reality is what no one at HP wants to hear: Lie down on the cold cotton sheets and enter corporate dotage, the future belongs to the generations that followed you.

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Whitman in interviews likes to ask how a company generating billions in cash and hundreds of billions in revenue could be falling behind. The answer lies in HP’s history. In 1999, after Lew Platt stepped down after an impressive six-and-a-half-year run (HP’s stock rose 700% vs. the Dow’s 247% gain), Carly Fiorina stepped in and bought Compaq to prepare HP for the 21st century. Only Fiorina’s new century wasn’t the one we live in now, where the PCs Compaq made are becoming less relevant.

Enter Mark Hurd. In 2005, he embarked on a series of aggressive purchases, a strategy that worked for several years. In April 2010, HP’s stock traded as high as $54.75 a share, thanks to Hurd’s campaign to remake HP into an IT-services powerhouse through multiple bold acquisitions.

Back in April 2010, HP’s stock was treading near $55 a share. Hurd had already bought EDS for $13.9 billion in 2008, following big-ticket deals like Opsware ($1.6 billion) and Mercury Interactive ($4.5 billion).

That month, Hurd closed one deal (3Com for $2.7 billion) and announced another (Palm for $1.2 billion). Hurd spent $24 billion on those major deals plus billions more on other, smaller buys. After he left, his pre-Whitman successors — notably Léo Apotheker — spent yet more on big-ticket companies: $2.35 billion on 3Par and, infamously, $11 billion on Autonomy.

Aggressive M&A can strengthen a company by broadening its reach into new markets. Or it can deplete its strength by distracting the company with complex IT integration, endless rounds of layoffs, and billions of dollars in write-downs. HP’s fate ended up being the latter. By November 2012, its stock had fallen to $11.35 a share, a decline of 79% over 25 months. During that same period, the S&P 500 rose 18%.

Whitman came in during the middle of that freefall, becoming HP’s CEO in 2011. She had her work cut out for her: more layoffs and writedowns, particularly in the wake of the controversial Autonomy deal. But starting last November, the stock rallied, suggesting that her efforts to steer HP out of the storm and into calmer waters was actually working.

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HP’s stock reached as high as $27.78 last month, an increase of 145% from its low point last fall. Then came its most recent financial report. The stock dropped 13% to $22 a share in a matter of hours after HP posted its earnings, largely because the numbers raised questions about whether this company could ever be turned around, even with the work that Whitman has done. The stock has stayed around $22 a share in the three weeks since then.

The reason has nothing to do with HP’s last quarter: Revenue and earnings came in line with Wall Street’s expectations, which were grim ones. PC revenue fell 11%, printer revenue (long HP’s cash cow) fell 4%, enterprise business fell 9%, and financial services fell 6%. The only bright spot was HP’s software, which saw revenue rise 1% to $982 million, or 3.6% of HP’s total revenue.

All that was expected. The fall of the old-school PC and the stagnant spending by companies on IT services have hurt HP and other tech giants. The killer was what HP saw ahead, which was not the stuff of recovery. In the conference call, Whitman said things like this:

“The server market growth rates have come down in the last quarter. The PC market has not stabilized as much as I had anticipated it would. That stabilization is yet to occur. Then finally, Enterprise Services, which is good for this year is the revenues running off more slowly this year …”

Analysts kind of freaked. Citigroup fretted that its “conference call unveiled several challenges.” Needham & Co., noting its earlier skepticism in a turnaround, said “euphoria is finally wearing off.” Others were were taken aback about the prospect of no revenue growth in 2014 and still others by the management changes in the enterprise division, an area that HP long hoped would be a growth area.

The past month has painted a picture where HP is not only a Silicon Valley pioneer that has fallen on hard times, but one where a turnaround may in fact be impossible even in the next few years. Given it employs well over 330,000 people, that’s a sobering scenario. HP will survive being jettisoned from the Dow. It may fare less well if it also slips from being an icon of Silicon Valley.