To apply the Distributive Property to simplify the expression \( (2+i)(4+5i) \), we can use the distributive property (also known as the FOIL method for binomials). The expression can be expanded as follows:
\[ (2+i)(4+5i) = 2 \cdot 4 + 2 \cdot 5i + i \cdot 4 + i \cdot 5i \]
Now, calculate each term:
- \( 2 \cdot 4 = 8 \)
- \( 2 \cdot 5i = 10i \)
- \( i \cdot 4 = 4i \)
- \( i \cdot 5i = 5i^2 \)
Since \( i^2 = -1 \), we can simplify \( 5i^2 \):
\[ 5i^2 = 5(-1) = -5 \]
Now, put all the calculated parts together:
\[ 8 + 10i + 4i - 5 \]
Combine the like terms (\( 8 - 5 \) and \( 10i + 4i \)):
\[ 8 - 5 = 3 \] \[ 10i + 4i = 14i \]
So, the final simplified expression is:
\[ 3 + 14i \]
Thus, the simplified expression is
\[ \boxed{3 + 14i} \]