Answer
$(t+10+2y)(t+10-2y)$
Work Step by Step
Grouping the first $3$ terms the given expression is equivalent to
\begin{array}{l}\require{cancel}
100+20t+t^2-4y^2
\\\\=
(100+20t+t^2)-4y^2
\\\\=
(t^2+20t+100)-4y^2
.\end{array}
Using the factoring of trinomials in the form $x^2+bx+c,$ the $\text{
expression
}$
\begin{array}{l}\require{cancel}
(t^2+20t+100)
\end{array} has $c=
100
$ and $b=
20
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
10,10
\right\}.$ Using these two numbers, the $\text{
expression
}$ above is equivalent to
\begin{array}{l}\require{cancel}
(t+10)(t+10)-4y^2
\\\\=
(t+10)^2-4y^2
.\end{array}
The expressions $
(t+10)^2
$ and $
4y^2
$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $
(t+10)^2-4y^2
,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(t+10)^2-(2y)^2
\\\\=
(t+10+2y)(t+10-2y)
.\end{array}
