Expand ${({x^2} + {y^2} + {z^2})^2}$

Hint: Here we are asked to expand the given expression. As we can see that the given expression is an algebraic expression with three unknown variables raised to the power two. The power that is outside the brackets can be expanded by multiplying the expression inside the brackets to itself. Then these two expressions can be multiplied and simplified if needed to get the expanded form of the given expression.

Complete step by step solution:
Since the product of the two values means, which is the process of using the multiplication operation only. Hence the multiplication of the ${({x^2} + {y^2} + {z^2})^2}$ is the requirement
Let $({x^2} + {y^2} + {z^2}) \times ({x^2} + {y^2} + {z^2})$ be the expression of the given question.
Since applying the multiplication we get
$
({x^2} + {y^2} + {z^2}) \times ({x^2} + {y^2} + {z^2}) \Rightarrow \\
({x^2} \times {x^2}) + ({x^2} \times {y^2}) + ({x^2} \times {z^2}) + ({y^2} \times {x^2}) + ({y^2} \times {y^2}) + ({y^2} \times {z^2}) + ({z^2} \times {x^2}) + ({z^2} \times {y^2}) + ({z^2} \times {z^2}) \\
$
Thus, using the simple addition and multiplication we have ${({x^2})^2} + {({y^2})^2} + {({z^2})^2} + 2({x^2})({y^2}) + 2({z^2})({y^2}) + 2({x^2})({z^2})$
Thus, we get ${x^4} + {y^4} + {z^4} + 2{x^2}{y^2} + 2{y^2}{z^2} + 2{z^2}{x^2}$
Now let us group the like terms and simplify them.
Hence, we get the product of $({x^2} + {y^2} + {z^2}) \times ({x^2} + {y^2} + {z^2})$ as ${x^4} + {y^4} + {z^4} + 2{x^2}{y^2} + 2{y^2}{z^2} + 2{z^2}{x^2}$

Note: As we know that while multiplying two algebraic expressions, we have to multiply it term by term and write it as a sum along with its sign since they are all integers. This process will be repeated for a number of terms in any one of the expressions that we took first. Then after multiplication we need to group the like terms to simplify them. Like terms are nothing but the terms having same variables with same degree and unlike terms are nothing but the terms having different variables. We have to remember that while grouping for simplification like terms is only grouped, we cannot group unlike terms.