Questions: Factor by grouping: x^2 y + 6 y + 8 x^2 + 48

Solution Steps

To factor by grouping, we need to rearrange and group the terms in such a way that we can factor out common factors from each group.

1. Rearrange the terms to group them in pairs.
2. Factor out the greatest common factor (GCF) from each pair.
3. Look for a common binomial factor in the resulting expression.

We start with the expression \( x^2 y + 6y + 8x^2 + 48 \). To factor by grouping, we can rearrange the terms as follows: \[ x^2 y + 8x^2 + 6y + 48 \]

Next, we group the terms into two pairs: \[ (x^2 y + 8x^2) + (6y + 48) \]

Now, we factor out the greatest common factor from each group:

• From the first group \( x^2 y + 8x^2 \), we can factor out \( x^2 \): \[ x^2(y + 8) \]
• From the second group \( 6y + 48 \), we can factor out \( 6 \): \[ 6(y + 8) \]

Now we can combine the factored groups: \[ x^2(y + 8) + 6(y + 8) \] We notice that \( (y + 8) \) is a common factor, so we can factor it out: \[ (y + 8)(x^2 + 6) \]

Final Answer