\(\displaystyle (j^t + t^2) + t - j = 10050 - 2600 \implies (j - t)(j + t) - (j - t) = (j - t)(j + t - 1) = 7450.\)

I agree that your factoring is valid and can be used to solve the problem by extensive trial and error.

But if you are going to assume whole numbers you can use the integer root theorem and get an answer very efficiently from the quartic.

\(\displaystyle t^4 - 5200t^2 + t + 6,749,950 = 0.\) I have no idea why this does not render properly.

Factor the constant term with the absolutely obvious factor of 10

\(\displaystyle 6,749,950 = 10 *674,995.\)

Factor again by the obvious factor of 5

\(\displaystyle 6,749,950 = 10 * 5 * 134999.\)

It is obvious virtually by inspection that 2, 5, 10, and 25 are too small. 25 squared is 625 which would make j almost 2000, and that squared is way bigger than 10050. So try t = 50.

\(\displaystyle 50^4 - 5200 * 50^2 + 50 = 6,250,000 - 50 * 2500 + 50 =\)

\(\displaystyle 6,250,050 - 13,000,000 = -\ 6,749,950.\)

50 works.

I'd be amazed if this problem was not found in a section that mentions either the rational or integer root theorems or both.