Sure! Let's go through the expression [tex]\( x^2(y + 1) - y - 1 \)[/tex] step by step.
### Step-by-Step Simplification
1. Expression breakdown:
[tex]\[
x^2(y + 1) - y - 1
\][/tex]
2. Distribute [tex]\( x^2 \)[/tex] inside the parentheses [tex]\( (y + 1) \)[/tex]:
[tex]\[
x^2 \cdot y + x^2 \cdot 1 - y - 1
\][/tex]
This simplifies to:
[tex]\[
x^2y + x^2 - y - 1
\][/tex]
3. Combine like terms (if any):
In this case, there are no like terms to combine. The terms [tex]\(x^2y\)[/tex], [tex]\(x^2\)[/tex], [tex]\(-y\)[/tex], and [tex]\(-1\)[/tex] are all different and cannot be further simplified by combining.
### Final Simplified Expression:
[tex]\[
x^2(y + 1) - y - 1
\][/tex]
or equivalently:
[tex]\[
x^2y + x^2 - y - 1
\][/tex]
This is the simplified form of the expression, which matches the answer.
### Explanation:
- The expression [tex]\( x^2(y + 1) - y - 1 \)[/tex] is already simplified.
- By distributing [tex]\( x^2 \)[/tex] over [tex]\( (y + 1) \)[/tex], we get [tex]\( x^2y + x^2 \)[/tex].
- The remaining [tex]\(-y - 1\)[/tex] terms are left as they are since there are no like terms to combine with [tex]\( x^2y \)[/tex] and [tex]\( x^2 \)[/tex].
Thus, the final expression stays as [tex]\( x^2(y + 1) - y - 1 \)[/tex] or in expanded form [tex]\( x^2y + x^2 - y - 1 \)[/tex].
