Divide \$(-2 - 5i)/ ( 4 + 5i)\$.

The given expression

\$(-2 - 5i)/ ( 4 + 5i)\$

Find the conjugate of the denominator:

Conjugate of (4 + 5i) = 4 - 5i

Multiply the numerator and the denominator by the conjugate of the denominator:

\$(-2 - 5i) / ( 4 + 5i) times (4 - 5i) /( 4 - 5i)\$

Since \$i^2 = - 1\$.

Calculate the numerator and plug the \$i^2 = -1\$ and then make it simplify:

\$(4 + 5i)(4 - 5i) = 4^2 - (5i)^2\$

\$ (a - b)(a + b) = a^2 - b^2\$

\$16 - 25(-1) = 16 + 25 = 41\$

Calculate the numerator and plug the \$i^2 = -1\$ and then make it simplify:

\$(- 2 - 5i)(4 - 5i)\$

⇒ \$(- 2 times 4)(- 2 times (- 5i))(- 5i times 4)+(- 5i times (- 5i))\$

⇒ \$- 8 10i - 20i 25i^2\$

⇒ - 8 - 10i + 25(−1)

⇒ - 33 - 10i

Combine the results:

\$ (- 33 - 10i)/(41)\$

\$ - (33)/ 41 - (10)/(41) i\$

So, the result of the division in standard form is \$(-2 - 5i)/ ( 4 + 5i) = - (33)/ 41 - (10)/(41) i\$.

It is incorrect because it has the wrong signs for both the real and imaginary parts It should be \$ -(33)/41 - (10)/(41) i\$, not \$ -(33)/41 + (10)/(41) i\$

\$ -(33)/41 + (10)/(41) i\$

The correct result should have a negative sign for both the real and imaginary parts. In this option, the real part contains a positive sign so it is wrong

\$ (33)/41 - (10)/(41) i\$

This choice is incorrect because the signs in the expression \$ (33)/41 + (10)/(41) i\$ are opposite

\$ (33)/41 + (10)/(41) i\$

It accurately calculates the division of complex numbers. So this choice is correct

\$ -(33)/41 - (10)/(41) i\$

Option

D

Can you summarize what you’ve understood in the above steps?