First, let's analyze and simplify the given expression step-by-step:
[tex]\[
\frac{4 m^2}{2 m + 3 n} + 12 m n + \frac{9 n^2}{2 m + 3 n}
\][/tex]
1. Find a common denominator:
The common denominator for the fractions involved is [tex]\(2m + 3n\)[/tex].
2. Rewrite each term with the common denominator:
[tex]\[
\frac{4 m^2}{2 m + 3 n} \quad \text{and} \quad \frac{9 n^2}{2 m + 3 n} \quad \text{already have the common denominator.}
\][/tex]
We will need to modify [tex]\(12mn\)[/tex] to have the same denominator:
[tex]\[
12 m n = \frac{12 m n (2 m + 3 n)}{2 m + 3 n}
\][/tex]
3. Combine the fractions:
Now we can combine all the terms under the common denominator [tex]\((2 m + 3 n)\)[/tex]:
[tex]\[
\frac{4 m^2 + 12 m n (2 m + 3 n) + 9 n^2}{2 m + 3 n}
\][/tex]
4. Simplify the numerator:
Simplify the expression inside the numerator:
[tex]\[
12 m n (2 m + 3 n) = 12 m n \cdot 2 m + 12 m n \cdot 3 n = 24 m^2 n + 36 m n^2
\][/tex]
Adding this back into our combined fraction, we get:
[tex]\[
\frac{4 m^2 + 24 m^2 n + 36 m n^2 + 9 n^2}{2 m + 3 n}
\][/tex]
5. Combine like terms:
[tex]\[
4 m^2 + 24 m^2 n + 36 m n^2 + 9 n^2
\][/tex]
Combine like terms to simplify the numerator:
The final simplified form is:
[tex]\[
\boxed{\frac{4 m^2 + 24 m^2 n + 36 m n^2 + 9 n^2}{2 m + 3 n}}
\][/tex]
