Answer
$-15t^3+5t^2-3t+1$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$ or the product of $2$ binomials, the given expression, $
(1-3t)(1+5t^2)
$, is equivalent to
\begin{array}{l}\require{cancel}
1(1)+1(5t^2)-3t(1)-3t(5t^2)
\\\\=
1+5t^2-3t-15t^3
\\\\=
-15t^3+5t^2-3t+1
.\end{array}