The product of 1 plus two instances imaginary unit and three plus 4 instances imaginary unit is the given mathematical expression on this math drawback.

$(1+2i)(3+4i)$

The product of $1$ plus $2i$ and $3$ plus $4i$ is a mathematical illustration for the multiplication of them. Subsequently, the above mathematical expression may be written as follows.

$=,,,$ $(1+2i) instances (3+4i)$

### Trick to keep away from confusion in multiplication

Multiplying the phrases within the advanced quantity $3$ plus $4i$ by one other advanced quantity $1$ plus $2i$ confuses some learners. So, let’s denote the advanced quantity $1$ plus $2i$ by a variable. On this drawback, the advanced quantity $1+2i$ is denoted by a variable $z$.

$=,,,$ $z instances (3+4i)$

### Multiply the phrases by their coefficient

The advanced $3+4i$ is a binomial and its phrases are multiplied by a variable $z$. So, the phrases $3$ and $4i$ may be multiplied by the variable $z$ as per the distributive property of multiplication over the addition.

$=,,,$ $z instances 3$ $+$ $z instances 4i$

The usage of variable $z$ is over. So, exchange the variable $z$ by its precise worth within the above mathematical expression.

$=,,,$ $(1+2i) instances 3$ $+$ $(1+2i) instances 4i$

In accordance with the commutative property of multiplication, the positions of the components may be modified in every time period.

$=,,,$ $3 instances (1+2i)$ $+$ $4i instances (1+2i)$

Now, use the distributive property yet another time in every time period to distribute the coefficient over the addition of the phrases.

$=,,,$ $3 instances 1$ $+$ $3 instances 2i$ $+$ $4i instances 1$ $+$ $4i instances 2i$

### Simplify the Mathematical expression

The advanced quantity $1$ plus $2i$ is multiplied by one other advanced quantity $3$ plus $4i$. The multiplication of them fashioned a mathematical expression. Now, it’s time to simplify the mathematical expression to seek out the product of the given advanced numbers $1$ plus $2i$ and $3$ plus $4i$.

$=,,,$ $3$ $+$ $6i$ $+$ $4i$ $+$ $8i^2$

Second and third phrases are like phrases within the above mathematical expression. Subsequently, add the like phrases $6i$ and $4i$ to seek out the sum of them.

$=,,,$ $3$ $+$ $10i$ $+$ $8i^2$

$=,,,$ $3$ $+$ $10i$ $+$ $8 instances i^2$

In accordance with the advanced numbers, the sq. of imaginary unit is unfavourable one.

$=,,,$ $3$ $+$ $10i$ $+$ $8 instances (-1)$

$=,,,$ $3$ $+$ $10i$ $-$ $8$

Now, use the commutative property to put in writing the phrases in an order for our comfort.

$=,,,$ $3$ $-$ $8$ $+$ $10i$

Have a look at the primary and second phrases. They’re numbers. So, subtract the quantity $8$ from quantity $3$ to seek out distinction of them.

$=,,,$ $-5+10i$