To simplify the given expression, we can combine like terms. The expression is:

$(5a^2 + 2b^2 + 3c^2) + (a^2 - b^2 + c^2)$

First, let's remove the parentheses (since there are no subtraction signs in front of them, this step simply involves rewriting the terms):

$5a^2 + 2b^2 + 3c^2 + a^2 - b^2 + c^2$

Now, let's combine like terms:

1. Combine the $a^2$ terms: $5a^2 + a^2 = 6a^2$
2. Combine the $b^2$ terms: $2b^2 - b^2 = b^2$
3. Combine the $c^2$ terms: $3c^2 + c^2 = 4c^2$

Thus, the simplified expression is:

$6a^2 + b^2 + 4c^2$

This is the result after combining like terms.

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Would you like further details or have any questions?

Here are 5 related questions that might interest you:

1. How do you identify and combine like terms in algebraic expressions?
2. What are some methods to factor quadratic expressions such as $6a^2 + b^2 + 4c^2$?
3. How would the expression change if there were coefficients with negative signs?
4. Can you solve an equation set to zero using the simplified form of this expression?
5. What are the rules for adding and subtracting algebraic expressions?

Tip: Always group like terms when simplifying expressions to keep your work organized and avoid mistakes.