Simplify the expression: ${(3x + 2y)^2} + {(3x - 2y)^2}$

Hint: Here we are asked to solve the given algebraic equation. As we can see that the given expression is sum of two expressions also those two expressions are raised to the power two so will first try to expand those two expressions by using the standard expansion of an expression like ${(a + b)^2} = (a + b)(a + b) \Rightarrow {a^2} + {b^2} + 2ab$ and ${(a - b)^2} = (a - b)(a - b) \Rightarrow {a^2} + {b^2} - 2ab$. Then we will simplify them to get the required answer.

Complete step by step solution:
Given the equation ${(3x + 2y)^2} + {(3x - 2y)^2}$ and then we need to find the value of the simplified form.
Since we know that ${(a + b)^2} = (a + b)(a + b) \Rightarrow {a^2} + {b^2} + 2ab$ and also ${(a - b)^2} = (a - b)(a - b) \Rightarrow {a^2} + {b^2} - 2ab$
Hence applying the same into the given variables, we get ${(3x + 2y)^2} = 9{x^2} + 12xy + 4{y^2}$ and ${(3x - 2y)^2} = 9{x^2} - 12xy + 4{y^2}$
Thus, using the addition and subtraction operation, we have ${(3x + 2y)^2} + {(3x - 2y)^2} = 9{x^2} + 12xy + 4{y^2} + (9{x^2} - 12xy + 4{y^2})$
Further solving we get ${(3x + 2y)^2} + {(3x - 2y)^2} = 9{x^2} + 12xy + 4{y^2} + 9{x^2} - 12xy + 4{y^2} \Rightarrow 9{x^2} + 9{x^2} + 4{y^2} + 4{y^2}$ and thus equals values with opposite signs cancel each other, thus we have ${(3x + 2y)^2} + {(3x - 2y)^2} = 18{x^2} + 8{y^2}$ which is the simplified form.

Note: In the above problem, after expanding those two expressions in the given question by using standard formulae we have simplified them, for that simplification, we need to have a knowledge about like and unlike terms. In simplifying the linear algebraic expressions, we cannot add or subtract them like normal numbers; we can only simplify them by grouping like terms together. Like terms are nothing but the terms having the same variables with the same degree and unlike terms are the terms having different variables. In the above problem, we can see that we have added the terms having the variable ${x^2}$ and ${y^2}$ with the terms having that same variable with the same power that is where we have grouped the terms for our simplification.