To factorize the expression [tex]\( p(a+b)^2 + 2(a+b) \)[/tex], we will take the following steps:
1. Identify common factors: First, we recognize that both terms in the expression contain the common factor [tex]\((a+b)\)[/tex].
2. Factor out the common term: We factor out [tex]\((a+b)\)[/tex] from each term in the expression.
Here's the detailed step-by-step factorization process:
### Step 1: Write out the expression
[tex]\[ p(a+b)^2 + 2(a+b) \][/tex]
### Step 2: Identify the common factor
Both terms contain [tex]\((a+b)\)[/tex] as a factor.
### Step 3: Factor out the common term [tex]\((a+b)\)[/tex]:
To understand this clearly, let's write the expression in a way that highlights the common factor:
[tex]\[ (a+b)(p(a+b)) + (a+b)(2) \][/tex]
Since [tex]\((a+b)\)[/tex] is present in both terms, we can factor it out:
[tex]\[ (a+b) \left[ p(a+b) + 2 \right] \][/tex]
### Step 4: Simplify inside the brackets
Now we simplify the term inside the brackets:
[tex]\[ p(a+b) + 2 = pa + pb + 2 \][/tex]
Putting it all together, the factored form is:
[tex]\[ (a+b)(pa + pb + 2) \][/tex]
Thus, the factorized expression is:
[tex]\[ (a+b)(a p + b p + 2) \][/tex]
This matches the final result:
[tex]\[ (a+b) (a p + b p + 2) \][/tex]
Therefore,
[tex]\[ p(a+b)^2 + 2(a+b) = (a+b)(a p + b p + 2). \][/tex]
