Answer

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**Hint:**We can easily solve problems on expanding squares by multiplying $\left( 5x+4y \right)$ with itself. Then by using the foil method where we multiply each of the individual terms in the left parenthesis by each individual term in the right parenthesis. Then we simplify and add the like terms to reach to the solution of the given product.

**Complete step by step answer:**

The expression we have is

${{\left( 5x+4y \right)}^{2}}$

If a term is squared or it has $2$ as its power value then we can say it is basically the multiplication of the term with itself.

So, we can rewrite the given expression as

$\Rightarrow \left( 5x+4y \right)$

Now, we apply the Foil method for multiplication in the above expression.

According to the Foil method if two terms are multiplied with each other then we should multiply each of the individual terms in the left parenthesis by each individual term in the right parenthesis.

Applying Foil method on the above expression we get

$\Rightarrow \left( 5x \right)\left( 5x \right)+\left( 5x \right)\left( 4y \right)+\left( 4y \right)\left( 5x \right)+\left( 4y \right)\left( 4y \right)$

Completing the multiplications of the above expression we get

$\Rightarrow 25{{x}^{2}}+20xy+20yx+16{{y}^{2}}$

We now combine the like terms of the above expression as shown below

$\Rightarrow 25{{x}^{2}}+20\left( xy+yx \right)+16{{y}^{2}}$

Adding the algebraic terms in the middle term of the above expression we get

$\Rightarrow 25{{x}^{2}}+20\left( 2xy \right)+16{{y}^{2}}$

Simplifying the above expression, we get

$\Rightarrow 25{{x}^{2}}+40xy+16{{y}^{2}}$

**Therefore, we conclude to the result of the product ${{\left( 5x+4y \right)}^{2}}$ as $25{{x}^{2}}+40xy+16{{y}^{2}}$.**

**Note:**While taking the negative and positive signs we must be careful so that they are properly taken into account. Also, while multiplying with the help of the foil method we must multiply the terms properly so that mistakes can be avoided. The given expression can also be solved by using the formula of ${{\left( a+b \right)}^{2}}$ , where $a=5x$ and $b=4y$ . We can substitute the above relations in the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ Thus, we get the result same as we have already got in our solution.

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