Answer
$x^4+18x^3+83x^2+18x+1$
Work Step by Step
Using $(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc$ or the square of a multinomial, then,
\begin{array}{l}
(x^2+9x+1)^2
\\=
(x^2)^2+(9x)^2+(1)^2+2(x^2)(9x)+2(x^2)(1)+2(9x)(1)
\\=
x^4+81x^2+1+18x^3+2x^2+18x
\\=
x^4+18x^3+(81x^2+2x^2)+18x+1
\\=
x^4+18x^3+83x^2+18x+1
.\end{array}
