Fractorise: `ab (x ^(2) + y^(2) ) + xy (a^(2) + b ^(2)).`

Options

A. ` (bx-ay) (ax+ by)`
B. ` (bx+ay) (ax+ by)`
C. `(bx+ay) (ax- by)`
D. `(bx-ay) (ax- by)`

Answer

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

### Step 1: Expand the expression
First, we will expand the expression to understand its components better.

\[ ab(x^2 + y^2) + xy(a^2 + b^2) = abx^2 + aby^2 + xya^2 + xyb^2 \]

### Step 2: Rearrange the terms
Next, we can rearrange the terms in a way that groups similar terms together.

\[ = abx^2 + xya^2 + xyb^2 + aby^2 \]

### Step 3: Group the terms
Now, we will group the terms in pairs to factor them more easily.

\[ = (abx^2 + aby^2) + (xya^2 + xyb^2) \]

### Step 4: Factor out common factors from each group
Now, we will factor out the common factors from each group.

From the first group \( abx^2 + aby^2 \), we can factor out \( ab \):

\[ = ab(x^2 + y^2) \]

From the second group \( xya^2 + xyb^2 \), we can factor out \( xy \):

\[ = xy(a^2 + b^2) \]

Putting it all together, we have:

\[ = ab(x^2 + y^2) + xy(a^2 + b^2) \]

### Step 5: Identify common factors
Now we can see that both parts of the expression share a common factor. We can factor out \( (bx + ya) \):

\[ = (bx + ya)(a + b) \]

### Final Factored Form
Thus, the final factored form of the expression is:

\[ = (bx + ya)(a + b) \]