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\$