Answer
$25x^2+10x+1+60xy+12y+36y^2$
Work Step by Step
Using $(a+b)^2=a^2+2ab+b^2$ or the square of a binomial, the given expression, $
[(5x+1)+6y]^2
,$ is equivalent to
\begin{array}{l}\require{cancel}
(5x+1)^2+2(5x+1)(6y)+(6y)^2
\\\\=
(5x+1)^2+12y(5x+1)+36y^2
\\\\=
(5x+1)^2+60xy+12y+36y^2
\\\\=
[(5x)^2+2(5x)(1)+(1)^2]+60xy+12y+36y^2
\\\\=
25x^2+10x+1+60xy+12y+36y^2
.\end{array}