To solve the expression \((2 - 5i)(p + q)(i)\) given that \( p = 2 \) and \( q = 5i \), follow these steps:
1. Substitute the values for \( p \) and \( q \):
Given:
[tex]\[
p = 2
\][/tex]
[tex]\[
q = 5i
\][/tex]
2. Form the second term:
Substitute \( p \) and \( q \) into \( p + q \):
[tex]\[
p + q = 2 + 5i
\][/tex]
3. Multiply the first term by the second term:
The expression becomes:
[tex]\[
(2 - 5i)(2 + 5i)
\][/tex]
To expand the product, use the distributive property (FOIL method):
[tex]\[
(2 - 5i)(2 + 5i) = 2 \cdot 2 + 2 \cdot 5i - 5i \cdot 2 - 5i \cdot 5i
\][/tex]
[tex]\[
= 4 + 10i - 10i - 25i^2
\][/tex]
Note that \( i^2 = -1 \):
[tex]\[
= 4 + 10i - 10i - 25(-1)
\][/tex]
[tex]\[
= 4 + 25
\][/tex]
[tex]\[
= 29
\][/tex]
4. Multiply the result by \( i \):
Now, take the result \( 29 \) and multiply it by \( i \):
[tex]\[
29 \cdot i = 29i
\][/tex]
Therefore, the evaluated expression \((2 - 5i)(2 + 5i)(i)\) is:
[tex]\[
\boxed{29i}
\][/tex]
This matches the true result described in the problem.
