How do you multiply \[\left( {7 + 3i} \right)\left( { - 7 + 5i} \right)\left( { - 8 - 4i} \right)\]?

Hint: In order to determine the multiplication of the above complex number, we have to first multiply the first two binomials in order to obtain a single binomial and again apply the multiplication of the remaining binomials and simplify it to obtain the required result.

Formula Used:
${i^2} = - 1$
${a^m} \times {a^n} = {a^{m + n}}$

Complete step-by-step solution:
The binomial concept will come under the topic of algebraic expressions. The algebraic expression is a combination of variables and constant. The alphabets are known as variables and the numerals are known as constants. In algebraic expression or equation, we have 3 types namely, monomial, binomial and polynomial.
A polynomial equation with two terms joined by the arithmetic operation + or – is called a binomial equation.
Now let us consider the given complex expression \[\left( {7 + 3i} \right)\left( { - 7 + 5i} \right)\left( { - 8 - 4i} \right)\]
Here we have three binomials as \[\left( {7 + 3i} \right),\left( { - 7 + 5i} \right)\]and\[\left( { - 8 - 4i} \right)\].
Here i is the imaginary number which is commonly known as the iota.
On the first we will multiply the first two binomial and simplify to convert them into single term
\[ \Rightarrow \left( {7\left( { - 7 + 5i} \right) + 3i\left( { - 7 + 5i} \right)} \right)\left( { - 8 - 4i} \right)\]
On multiplying we get
\[ \Rightarrow \left( { - 49 + 35i - 21i + 15{i^2}} \right)\left( { - 8 - 4i} \right)\]
Simplifying the above further by combining like terms and putting ${i^2} = - 1$
\[
   \Rightarrow \left( { - 49 + 14i + 15\left( { - 1} \right)} \right)\left( { - 8 - 4i} \right) \\
   \Rightarrow \left( { - 49 + 14i - 15} \right)\left( { - 8 - 4i} \right) \\
   \Rightarrow \left( { - 64 + 14i} \right)\left( { - 8 - 4i} \right) \\
 \]
Now we got two binomials,
Repeating the same procedure of multiplication of binomials, we get
\[
   \Rightarrow - 64\left( { - 8 - 4i} \right) + 14i\left( { - 8 - 4i} \right) \\
   \Rightarrow 512 + 256i - 112i - 56{i^2} \\
 \]
Simplifying the above further by combining like terms and putting ${i^2} = - 1$
\[
   \Rightarrow 512 + 256i - 112i - 56\left( { - 1} \right) \\
   \Rightarrow 512 + 144i + 56 \\
   \Rightarrow 568 + 144i \\
 \]

Therefore, the multiplication of the given expression is \[568 + 144i\].

Additional information:
1. Real Number: Any number which is available in a number system, for example, positive, negative, zero, whole number, discerning, unreasonable, parts, and so forth are Real numbers. For instance: $12, - 45, 0, 1/7, 2.8, \sqrt{5}$ and so forth, are all the real numbers.
2. A Complex number is a number which are expressed in the form $a + ib$ where $ib$ is the imaginary part and $a$ is the real number .i is generally known by the name iota. \[\]
or in simple words complex numbers are the combination of a real number and an imaginary number .
3.The Addition or multiplication of any 2-conjugate complex number always gives an answer which is a real number.

Note:
1. Complex numbers are very useful in representing periodic motion like water waves, light waves and current and many more things which depend on sine or cosine waves.
2. Complex conjugate of $a + ib$ is $a - ib$
3. ${i^3}$is equal to $ - i$ as ${i^3} = i.{i^2} = i.\left( { - 1} \right) = - i$
4. Like terms are the terms having the same variable and power.
5. Make sure all the like terms combine properly for the simplification.