You do not qualify because Scholarships, grants, and other tax free assistance exceed the education expenses.
But, as TeaxaRoger suggested, there is a tax “loophole” available. The student reports all his scholarship, up to the amount needed to claim the American opportunity credit (AOTC), as income on his return. That way, the parents (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship. You cannot do this if the school’s billing statement specifically shows the scholarships being applied to tuition or if the conditions of the grant are that it be used to pay for qualified expenses.
Using your numbers as an example: Student has $4258 in box 5 of the 1098-T and $1775 in box 1, and $400 of other qualified expenses. At first glance she has $2083 of taxable income and nobody can claim the American opportunity credit. But if she reports the full $4258 as income on her return, the parents can claim $2175 of qualified expenses on their return, for the AOTC.
In actuality, she does not even need to file, because her total income is less than the $12,200 filing requirement*. Some experts, recommend that she file anyway, to document that she treated the scholarship as taxable income.
You can both use the 1098-T to enter the expenses. If you claim the tuition credit, you do need to report that you got one (the TurboTax interview will handle this) Your student should use the 1098-T because it makes entering scholarship income go smoother.
You essentially should use a work around in TurboTax (TT). Here's how I would do it. Enter the 1098-T, on your return, but leave box 5 blank. Lying to TurboTax to get it to do what you want does not constitute lying to the IRS. Be sure to enter the book cost at the book screen (or add it to box 1 as an expedient).
On the student's return enter $0 box 1 and $4258 in box 5. Theoretically, the Turbotax interview can handle entering actual numbers. But it gets tricky. The key being you should eventually reach a screen called "Amount used to calculate education deduction or credit". I recommend the work around.
*For others reading this, the $12,200 filing requirement assumes the student only has earned income and/or scholarship income and not other kinds of income (including taxable 529 plan distributions).
