The expression provided in the image is:

$x^2 + 4y^2 + 25z^2 - 4xy - 20yz - 10xz$

We are given the values $x = 4$, $y = 2$, and $z = 3$. Let’s substitute these values into the expression and simplify.

$(4)^2 + 4(2)^2 + 25(3)^2 - 4(4)(2) - 20(2)(3) - 10(4)(3)$

Now, I will calculate the result step by step.

$16 + 4(4) + 25(9) - 4(8) - 20(6) - 10(12)$

Let me calculate this.The value of the expression when $x = 4$, $y = 2$, and $z = 3$ is $-15$.

Would you like further clarification or details on any step?

Here are 5 related questions you could explore:

1. What happens if we change the value of $z$ in the given expression?
2. How does the expression simplify if the $xy$, $yz$, and $xz$ terms are removed?
3. What would the value of the expression be if $x = y = z = 1$?
4. Can we generalize this expression for other values of $x$, $y$, and $z$?
5. What role do the coefficients of the variables play in determining the result?

Tip: Always substitute values carefully and follow the order of operations (PEMDAS) to avoid errors in multi-variable expressions.