How do you simplify the expression 2 divided by the quantity of 2 plus 5i?

To simplify the expression 2 ÷ (2 + 5i), we can multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 2 + 5i is 2 – 5i.

By doing this, we can eliminate the imaginary part from the denominator:

   2          2 - 5i
-----  ×  -----------
2 + 5i      2 - 5i

This gives us:

      2(2 - 5i)
--------------------
(2 + 5i)(2 - 5i)

Now, let’s calculate the denominator:

(2 + 5i)(2 - 5i) = 2² - (5i)²
                  = 4 - 25(-1)
                  = 4 + 25
                  = 29

Next, for the numerator:

2(2 - 5i) = 4 - 10i

Putting it all together, we receive:

   4 - 10i
----------
    29

So, the simplified form of the original expression 2 ÷ (2 + 5i) is:

   4 - 10i
----------
    29

This can also be expressed as:

4/29 - (10/29)i

In conclusion, the simplified expression is:

  4/29 - (10/29)i