How do you multiply ${{\left( 5-8y \right)}^{2}}$ ?

Hint: In the given question we have to find the multiplication of the given expression, which can be done by writing the expression twice separately and then multiplying each variable and coefficients with the next term to get the required answer.

Complete step by step solution:
Given expression to multiply: ${{\left( 5-8y \right)}^{2}}$
First, we can expand the expression and write it as the product of two similar terms, that is,
$\Rightarrow {{\left( 5-8y \right)}^{2}}=\left( 5-8y \right)\left( 5-8y \right)$
Now, to multiply the above expression and to get the final answer to multiply each term of the second bracket with each term in the first bracket as shown below,
$\Rightarrow \left( 5-8y \right)\left( 5-8y \right)=5\left( 5-8y \right)-8y\left( 5-8y \right)$
Simplify the above expression by multiplying the terms outside the parentheses with terms inside the parenthesis to get,
$\Rightarrow 5\left( 5-8y \right)-8y\left( 5-8y \right)=\left( 5\times 5 \right)+\left( 5\times -8y \right)+\left( -8y\times 5 \right)+\left( -8y\times -8y \right)$
Now, carefully multiply all the terms together while taking care of the signs associated with each term to get the expression shown below,
$\Rightarrow \left( 5\times 5 \right)+\left( 5\times -8y \right)+\left( -8y\times 5 \right)+\left( -8y\times -8y \right)=25+\left( -40y \right)+\left( -40y \right)+64{{y}^{2}}$
Open the brackets of the terms with signs and write all the terms with their proper signs in front of them to get the following expression,
$\Rightarrow \left( 5\times 5 \right)+\left( 5\times -8y \right)+\left( -8y\times 5 \right)+\left( -8y\times -8y \right)=25+\left( -40y \right)+\left( -40y \right)+64{{y}^{2}}$
Now, add and subtract all the like terms, that is terms with variables together and the constants together to simplify the above expression and rewrite the terms in the standard form, to get the expression as shown below,
$\Rightarrow 25+\left( -40y \right)+\left( -40y \right)+64{{y}^{2}}=64{{y}^{2}}-80y+25$
Therefore, on multiplying and simplifying the given expression ${{\left( 5-8y \right)}^{2}}$ we get the final answer as $64{{y}^{2}}-80y+25$.

Note: While simplifying the expression after opening the parenthesis, extra care and attention should be given to the signs that are associated with each term. Also, while adding and subtracting the like terms, care should be taken to keep the signs of the terms clear to avoid any miscalculations.