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 correct option is 1600.

To simplify the given expression efficiently without performing lengthy multiplications, we can use an algebraic identity.

Let us observe the given expression:

$25\times 25+9\times 9+6\times 6+2\times 25\times 9+2\times 9\times 6+2\times 25\times 6$

This can be written in exponent form as:

${25}^2+9^2+6^2+2\left(25\right)\left(9\right)+2\left(9\right)\left(6\right)+2\left(25\right)\left(6\right)$

Notice that this matches the standard algebraic identity for the square of a trinomial:

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

By comparing the terms, we can let:
$a=25$
$b=9$
$c=6$

Now, substituting these values into the left side of the algebraic identity:

$\text{Expression}={(25+9+6)}^2$

First, perform the addition inside the parentheses:

$25+9+6=40$

Now, take the square of 40:

${40}^2=40\times 40=1600$

Thus, the simplified value of the given expression is 1600.