Question

using the properties of complex numbers, what is the result of the calculation $(2 + 3i)(4 - i) + (1 + 5i)$?
(1 point)

the result is the number $12 + 19i$.

the result is the number $6 + 15i$.

the result is the number $12 + 15i$.

the result is the number $11 + 10i$.

Explanation:

Step1: Multiply the complex numbers

First, we multiply \((2 + 3i)(4 - i)\) using the distributive property (FOIL method).

$$
LATEXBLOCK0
$$

Since \(i^2=-1\), we substitute that in:

$$
LATEXBLOCK1
$$

Step2: Add the remaining complex number

Now we add \((1 + 5i)\) to the result from Step 1:

$$
LATEXBLOCK2
$$

Wait, there must be a mistake. Let's re - calculate Step 1:

$$
LATEXBLOCK3
$$

Then add \((1 + 5i)\):

$$
LATEXBLOCK4
$$

Wait, but the first option is \(12 + 19i\), maybe I made a mistake. Let's re - do the multiplication:

$$
LATEXBLOCK5
$$

Adding \((1 + 5i)\): \(11+1 + 10i+5i=12 + 15i\). But the first option is \(12 + 19i\). Let's check the multiplication again:

$$
LATEXBLOCK6
$$

Adding \((1 + 5i)\): \(11 + 1=12\), \(10i+5i = 15i\). So the result is \(12 + 15i\). But the first option says \(12+19i\). There must be an error in my calculation. Wait, maybe the original problem was \((2 + 3i)(4 - i)+(1 + 9i)\)? No, the problem is \((2 + 3i)(4 - i)+(1 + 5i)\).

Wait, let's check the multiplication once more:

\(2\times4 = 8\), \(2\times(-i)=-2i\), \(3i\times4 = 12i\), \(3i\times(-i)=-3i^{2}\). Since \(i^{2}=-1\), \(-3i^{2}=3\). So \(8-2i + 12i+3=11 + 10i\). Then adding \(1 + 5i\): \(11+1=12\), \(10i + 5i=15i\). So the correct result is \(12 + 15i\). But the first option is \(12 + 19i\), maybe there is a typo in the problem or in my understanding.

Wait, maybe the problem is \((2 + 3i)(4 - i)+(1 + 9i)\). Let's try that:

After Step 1: \(11 + 10i\), adding \(1+9i\): \(11 + 1=12\), \(10i+9i = 19i\). So maybe there was a typo in the problem's \(1 + 5i\) and it should be \(1 + 9i\). But according to the given problem, if we follow the numbers:

Wait, let's re - calculate the multiplication:

\((2 + 3i)(4 - i)=2\times4-2i+3i\times4-3i\times i=8-2i + 12i-3i^{2}=8 + 10i+3=11 + 10i\)

Then add \((1 + 5i)\): \(11+1=12\), \(10i + 5i=15i\). So the result is \(12 + 15i\). But the first option is \(12 + 19i\), the second is \(6 + 15i\), the third is \(12 + 15i\), the fourth is \(11+10i\). So the correct answer should be the third option: "The result is the number \(12 + 15i\)". But in my initial wrong calculation, I thought I made a mistake, but now I see.

Wait, the options are:

1. The result is the number \(12 + 19i\).

1. The result is the number \(6 + 15i\).

1. The result is the number \(12 + 15i\).

1. The result is the number \(11 + 10i\).

So after correct calculation, the result is \(12 + 15i\), which is the third option.

Answer:

The result is the number \(12 + 15i\) (the third option: The result is the number \(12 + 15i\)).