Simplify the following expression:
$$25\times 25+9\times 9+6\times 6+2\times 25\times 9+2\times 9\times 6+2\times 25\times 6$$

The given expression is:
$$25\times 25+9\times 9+6\times 6+2\times 25\times 9+2\times 9\times 6+2\times 25\times 6$$

This can be rewritten in exponential terms as:

$$25^2+9^2+6^2+2(25)(9)+2(9)(6)+2(25)(6)$$

We recognize that this fits the standard algebraic expansion identity:

$$(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca$$

By comparing the terms, we can define:

$$a=25,b=9,c=6$$

Substituting these values into the factored identity form:

$$(25+9+6{)}^2=(40{)}^2$$

$$(40{)}^2=1600$$

Therefore, the simplified value of the expression is $1600$.