[tex]f(t)=3t^2+5t-7\implies f(5)=3(5)^2+5(5)-7=93[/tex]
which means the remainder upon dividing [tex]f(t)[/tex] by [tex]t-5[/tex] is 93.
This is because we can write
[tex]\dfrac{f(t)}{t-5}=q(t)+\dfrac{r(t)}{t-5}[/tex]
where [tex]q(t)[/tex] is the quotient and [tex]r(t)[/tex] is the remainder. Multiplying both sides by [tex]t-5[/tex], then evaluating at [tex]t=5[/tex] makes the [tex]q(t)[/tex] term vanish, leaving the remainder:
[tex]f(t)=q(t)(t-5)+r(t)\implies f(5)=r(5)[/tex]
