Question

Factorise the given equation:-\[{p^2} + 6p + 8\]
A.\[\left( {p + 2} \right)\left( {p - 2} \right)\]
B.\[\left( {p + 4} \right)\left( {2p - 2} \right)\]
C.\[\left( {2p + 4} \right)\left( {p + 2} \right)\]
D.\[\left( {p + 4} \right)\left( {p + 2} \right)\]

Answer 1

Hint: Here, we have to use the concept of the factorisation. Factorisation is the process in which a number is written in the forms of its small factors which on multiplication give the original number.

Complete step by step solution:
We will first split the middle term of the equation \[{p^2} + 6p + 8\] into two parts such that its multiplication will be equal to the product of the first term and the third term of the equation. Therefore, we get
\[ \Rightarrow {p^2} + 6p + 8 = {p^2} + 4p + 2p + 8\]
Now we will be taking \[p\] common from the first two terms and taking 2 common from the last two terms. Therefore the equation becomes
\[ \Rightarrow {p^2} + 6p + 8 = p\left( {p + 4} \right) + 2\left( {p + 4} \right)\]
Now factoring out \[\left( {p + 4} \right)\] from the equation, we get
\[ \Rightarrow {p^2} + 6p + 8 = \left( {p + 4} \right)\left( {p + 2} \right)\]
Hence, \[\left( {p + 4} \right)\left( {p + 2} \right)\] is the factor of the given equation \[{p^2} + 6p + 8\].
So, option D is the correct option.

Note: Here we have to split the middle term very carefully and according to the basic condition which is that the middle term i.e. term with the single power of the variable should be divided in such a way that its multiplication must be equal to the product of the first and the last term of the equation. Factors can be the same but in our case it’s different. We should know that the factors we have obtained on solving that we will get the original given equation. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations. Generally in these types of questions, algebraic identities are used to solve and make the factors.