Let's solve the given expression step-by-step. The expression is:
[tex]\[
\left(\frac{(\sqrt{m}+n)^2}{2}\right) (3 p)
\][/tex]
Given the values [tex]\( m = 4 \)[/tex], [tex]\( n = 2 \)[/tex], and [tex]\( p = 3 \)[/tex]:
1. Calculate the square root of [tex]\( m \)[/tex]:
[tex]\[
\sqrt{m} = \sqrt{4} = 2
\][/tex]
2. Add [tex]\( n \)[/tex] to the square root of [tex]\( m \)[/tex]:
[tex]\[
\sqrt{m} + n = 2 + 2 = 4
\][/tex]
3. Square the resulting sum:
[tex]\[
(\sqrt{m} + n)^2 = 4^2 = 16
\][/tex]
4. Divide the squared result by 2:
[tex]\[
\frac{(\sqrt{m} + n)^2}{2} = \frac{16}{2} = 8
\][/tex]
5. Multiply the result by [tex]\( 3p \)[/tex]:
[tex]\[
\left( \frac{(\sqrt{m} + n)^2}{2} \right)(3p) = 8 \cdot (3 \times 3) = 8 \cdot 9 = 72
\][/tex]
Thus, the final result of the given expression is:
[tex]\[
72
\][/tex]
Additionally, the intermediate results are:
[tex]\[
\sqrt{m} + n = 4, \quad (\sqrt{m} + n)^2 = 16, \quad \frac{(\sqrt{m} + n)^2}{2} = 8, \quad \text{and} \quad \left(\frac{(\sqrt{m}+n)^2}{2}\right)(3p) = 72.
\][/tex]
