we can start by looking at the leading terms:

t x ? = t^{4}.

clearly we need t^{3}, so our first "go-round" is to multiply t - 1 by t^{3}, and subtract the result from t^{4} - 3tq + t + 5.

note we can re-write this as:

t^{4} + 0t^{3} + 0t^{2} + (1 - 3q)t + 5.

now (t - 1)(t^{3}) = t^{4} - t^{3}, and:

t^{4} + 0t^{3} + 0t^{2} + (1 - 3q)t + 5 - (t^{4} - t^{3}) = t^{3} + 0t^{2} + (1 - 3q)t + 5

we've "reduced the degree by one".

again, just looking at the leading terms we see we need to multiply t - 1 by t^{2}, and subtract, so (t - 1)t^{2} = t^{3} - t^{2}, and:

t^{3} + 0t^{2} + (1 - 3q)t + 5 - (t^{3} - t^{2}) = t^{2} + (1 - 3q)t + 5

now we need to multiply t - 1 by t, and subtract: (t - 1)t = t^{2} - t, and:

t^{2} + (1 - 3q)t + 5 - (t^{2} - t) = (2 - 3q)t + 5

here, we need to multiply by 2 - 3q (this will be the "last go-round"), and (t - 1)(2 - 3q) = (2 - 3q)t - (2 - 3q). another subtraction:

(2 - 3q)t + 5 - ((2 - 3q)t - (2 - 3q)) = 5 + (2 - 3q) = 7 - 3q.

now, without knowing what "q" is, this is as far as we can go.

what we've done is shown that:

t^{4} - 3tq + t + 5 = (t^{3} + t^{2} + t + (2 - 3q))(t - 1) + 7 - 3q.

the quotient is: t^{3} + t^{2} + t + (2 - 3q)

the remainder is: 7 - 3q.