factorize : `ab(x^2+1)+x(a^2+b^2)`

Answer

To factorize the expression \( ab(x^2 + 1) + x(a^2 + b^2) \), we can follow these steps:

### Step 1: Rewrite the expression
Start with the given expression:
\[ ab(x^2 + 1) + x(a^2 + b^2) \]

### Step 2: Expand the expression
Distributing \( ab \) and \( x \):
\[ = abx^2 + ab + xa^2 + xb^2 \]

### Step 3: Group the terms
Now, we can group the terms in pairs:
\[ = (abx^2 + xa^2) + (ab + xb^2) \]

### Step 4: Factor out common terms
From the first group \( abx^2 + xa^2 \), we can factor out \( x \):
\[ = x(abx + a) + (ab + xb^2) \]
From the second group \( ab + xb^2 \), we can factor out \( b \):
\[ = x(abx + a) + b(a + bx) \]

### Step 5: Look for common factors
Now, we can see that both terms have a common factor of \( (a + bx) \):
\[ = (a + bx)(x + b) \]

### Final Factorized Form
Thus, the factorized form of the expression is:
\[ = (a + bx)(x + b) \]