Which of the following values of $x$ satisfies the equation $81^{4x+4}=243^{2x+8}$?

Given equation:
$$81^{4x+4}=243^{2x+8}$$
Express both bases as powers of $3$:
$$81=3^4$$
$$243=3^5$$
Substitute these into the equation:
$$(3^4{)}^{4x+4}=(3^5{)}^{2x+8}$$
Apply the power law $(a^m{)}^n=a^{m\cdot n}$:
$$3^{4(4x+4)}=3^{5(2x+8)}$$
$$3^{16x+16}=3^{10x+40}$$
Since the bases are identical, equate the exponents:
$$16x+16=10x+40$$
$$16x-10x=40-16$$
$$6x=24$$
$$x=4$$