Factorise: `ab (c ^(2) +1) + c (a ^(2) + b ^(2))`

Options

A. `(ab + c) (a +bc)`
B. `(ac + b) ( ab + c)`
C. `(a+ bc)(ac+b)`
D. `(a+b) (ac +b)`

Answer

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

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

### Step 2: Expand the expression
We can expand the expression to see if we can find common terms:
\[ = abc^2 + ab + ca^2 + cb^2 \]

### Step 3: Group the terms
Now, we can group the terms in a way that allows us to factor them:
\[ = abc^2 + ca^2 + cb^2 + ab \]
We can group the first two terms and the last two terms:
\[ = (abc^2 + ca^2) + (cb^2 + ab) \]

### Step 4: Factor out common terms
Now we can factor out common factors from each group:
1. From the first group \( abc^2 + ca^2 \), we can factor out \( ac \):
\[ ac(bc + a) \]
2. From the second group \( cb^2 + ab \), we can factor out \( b \):
\[ b(c + a) \]

Putting it all together, we have:
\[ = ac(bc + a) + b(c + a) \]

### Step 5: Factor out the common binomial
Now we can see that \( (bc + a) \) and \( (c + a) \) can be factored out:
\[ = (ac + b)(bc + a) \]

### Final Answer
Thus, the factorized form of the expression is:
\[ (ac + b)(bc + a) \]

---