Factorize x2 + 2a + ax + 2x.

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, we’ll walk through a step-by-step 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, let’s move on to the detailed step-by-step explanation.

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Step-by-Step Factorization

1. Identify the Expression: The given expression is:
   x2+2a+ax+2xx^2 + 2a + ax + 2xx2+2a+ax+2x
2. Rearrange the Terms: To make it easier to factor, let’s rearrange the terms to group similar variables:
   x2+ax+2x+2ax^2 + ax + 2x + 2ax2+ax+2x+2a
3. 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:

1. • 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)
2. So the expression now looks like:
   x(x+a)+2(x+a)x(x + a) + 2(x + a)x(x+a)+2(x+a)
3. 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:

1. 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

 

1. 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:

1. Rearrange the expression to group similar terms.
2. Group the terms and factor out the common factors in each group.
3. 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 step-by-step explanation helps you understand the factorization process! If you have any more questions, feel free to ask.