Question

How do you multiply \[\left( {4n + 1} \right)\left( {2n + 6} \right)\] ?

Answer 1

Hint: This problem is related to basic expansion formulas. These are used in algebraic expression frequently. Here we are given terms of the form \[\left( {a + b} \right)\left( {c + d} \right)\] . So initially we will multiply the first term of first bracket \[\left( {4n} \right)\] with both the terms of second bracket and then multiply second term of first bracket \[\left( 1 \right)\] with both terms of second bracket. If there are any terms that can be solved so we will simplify them. Try to write them in standard form.

Complete step-by-step answer:
Given that \[\left( {4n + 1} \right)\left( {2n + 6} \right)\]
So let’s multiply to simplify the expression.
Here n is a variable only. So initially we will multiply the first term of the first bracket \[\left( {4n} \right)\] with both the terms of the second bracket.
 \[ \Rightarrow 4n\left( {2n + 6} \right)\]
And now multiply second term of first bracket \[\left( 1 \right)\] with both terms of second bracket
 \[ \Rightarrow 1\left( {2n + 6} \right)\]
On combining we get the actual expression
 \[ \Rightarrow 4n\left( {2n + 6} \right) + 1\left( {2n + 6} \right)\]
Now we will multiply the terms
 \[ \Rightarrow 4n \times 2n + 4n \times 6 + 1 \times 2n + 1 \times 6\]
On separating constants and variables
 \[ \Rightarrow 4 \times 2 \times n \times n + 4 \times 6 \times n + 1 \times 2n + 1 \times 6\]
Any number if multiplied with it gives the square,
 \[ \Rightarrow 4 \times 2 \times {n^2} + 4 \times 6 \times n + 1 \times 2n + 1 \times 6\]
Now multiply the constants,
 \[ \Rightarrow 8 \times {n^2} + 24 \times n + 2n + 6\]
Remove the multiplication sign,
 \[ \Rightarrow 8{n^2} + 24n + 2n + 6\]
Here in second and third term has same variable as coefficient so we can add them,
 \[ \Rightarrow 8{n^2} + 26n + 6\]
This is our final answer.
So, the correct answer is “ \[ 8{n^2} + 26n + 6\] ”.

Note: Note that whenever we solve these types of problems the sequence is very important because if we miss it the answer changes accordingly. Also note that when we add two terms their degree of coefficient should be the same. For example we cannot add two terms like \[8{n^2}\] and \[26n\] because both do not have the same degree coefficient. So remember that. Also don’t forget to check the signs of the terms. When we multiply the second terms of the first bracket we take it along with its sign. Like \[ \Rightarrow 4n\left( {2n + 6} \right) + 1\left( {2n + 6} \right)\] .
In these types of problems be careful about the signs and brackets. As in addition as multiplications of two signs it matters so much. Note the table given below.

signs	addition	multiplication
\[ + , + \]	\[ + \]	\[ + \]
\[ + , - \]	\[ + \] (first number is greater)	\[ - \]
\[ - , + \]	\[ - \] (negative number is greater)	\[ - \]
\[ - , - \]	\[ - \]	\[ + \]