To square the given expression \( (4m^2 + 19mn + 12n) \), we use the formula for squaring a trinomial:

\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
\]

Here, \( a = 4m^2 \), \( b = 19mn \), and \( c = 12n \). Let's break this down step-by-step:

1. **Square each term:**
   \[
   a^2 = (4m^2)^2 = 16m^4
   \]
   \[
   b^2 = (19mn)^2 = 361m^2n^2
   \]
   \[
   c^2 = (12n)^2 = 144n^2
   \]

2. **Calculate the product of each pair and multiply by 2:**
   \[
   2ab = 2 \times (4m^2) \times (19mn) = 2 \times 76m^3n = 152m^3n
   \]
   \[
   2bc = 2 \times (19mn) \times (12n) = 2 \times 228mn^2 = 456mn^2
   \]
   \[
   2ca = 2 \times (12n) \times (4m^2) = 2 \times 48m^2n = 96m^2n
   \]

3. **Combine all terms:**
   \[
   (4m^2 + 19mn + 12n)^2 = 16m^4 + 361m^2n^2 + 144n^2 + 152m^3n + 456mn^2 + 96m^2n
   \]

So, the expanded form of \( (4m^2 + 19mn + 12n)^2 \) is:

\[
16m^4 + 152m^3n + 457m^2n^2 + 456mn^2 + 144n^2
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you simplify expressions with higher powers?
2. What are the general steps to square a trinomial?
3. How do you multiply expressions containing multiple variables?
4. What is the difference between expanding and factoring polynomials?
5. How would you square a binomial, such as \( (a + b)^2 \)?

### Tip:
When squaring a trinomial, make sure to include all cross-product terms and double them to get the correct expansion.

The query submitted by the user: (4m^2 + 19mn + 12n)^2

Learn how to square the trinomial expression (4m^2 + 19mn + 12n)^2. This detailed solution explains the steps to expand the expression using the trinomial expansion formula and includes all cross-product terms. Suitable for students in Grades 9-12 studying algebra and polynomial operations.

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