Factorization is a process in algebra where we break down a complex expression into simpler components, known as factors, that when multiplied together give the original expression. This technique is useful in simplifying expressions and solving equations. Today, weâ€™ll walk through a stepbystep guide to factorize the expression x2+2a+ax+2xx^2 + 2a + ax + 2xx2+2a+ax+2x, making it easier to handle and understand.
Now that we have a clear and concise introduction, letâ€™s move on to the detailed stepbystep explanation.
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StepbyStep Factorization
 Identify the Expression: The given expression is:
x2+2a+ax+2xx^2 + 2a + ax + 2xx2+2a+ax+2x  Rearrange the Terms: To make it easier to factor, letâ€™s rearrange the terms to group similar variables:
x2+ax+2x+2ax^2 + ax + 2x + 2ax2+ax+2x+2a  Group the Terms: Group the terms in pairs to factor by grouping:
(x2+ax)+(2x+2a)(x^2 + ax) + (2x + 2a)(x2+ax)+(2x+2a)
Factor Out the Common Factors in Each Group:

 In the first group (x2+ax)(x^2 + ax)(x2+ax), factor out xxx: x(x+a)x(x + a)x(x+a)
 In the second group (2x+2a)(2x + 2a)(2x+2a), factor out 222: 2(x+a)2(x + a)2(x+a)
 So the expression now looks like:
x(x+a)+2(x+a)x(x + a) + 2(x + a)x(x+a)+2(x+a)  Factor Out the Common Binomial Factor: Notice that (x+a)(x + a)(x+a) is a common factor in both groups:
(x+a)(x+2)(x + a)(x + 2)(x+a)(x+2)
Final Factored Form
The factored form of the expression x2+2a+ax+2xx^2 + 2a + ax + 2xx2+2a+ax+2x is:
(x+a)(x+2)(x + a)(x + 2)(x+a)(x+2)
Verification
To verify, we can expand the factored form back to the original expression:
 Expand:
(x+a)(x+2)=x(x+2)+a(x+2)(x + a)(x + 2) = x(x + 2) + a(x + 2)(x+a)(x+2)=x(x+2)+a(x+2) =x2+2x+ax+2a= x^2 + 2x + ax + 2a=x2+2x+ax+2a
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 Combine Like Terms:
x2+2x+ax+2a=x2+ax+2x+2ax^2 + 2x + ax + 2a = x^2 + ax + 2x + 2ax2+2x+ax+2a=x2+ax+2x+2a
This matches our original expression, confirming that the factorization is correct.
Summary
In summary, to factorize x2+2a+ax+2xx^2 + 2a + ax + 2xx2+2a+ax+2x:
 Rearrange the expression to group similar terms.
 Group the terms and factor out the common factors in each group.
 Factor out the common binomial factor.
The final factored form is:
(x+a)(x+2)(x + a)(x + 2)(x+a)(x+2)
I hope this stepbystep explanation helps you understand the factorization process! If you have any more questions, feel free to ask.