Sure, let's break down and simplify each of the given expressions step by step. We aim to identify their simplest forms and then find any differences among them.
### Expression 1:
[tex]\[
\frac{x}{x^2 + 3x + 2} - \frac{1}{(x+2)(x+1)}
\][/tex]
1. Recognize that \(x^2 + 3x + 2\) factors as \((x + 2)(x + 1)\).
2. Rewrite the expression as:
[tex]\[
\frac{x}{(x+2)(x+1)} - \frac{1}{(x+2)(x+1)}
\][/tex]
3. Combine the fractions:
[tex]\[
\frac{x - 1}{(x+2)(x+1)}
\][/tex]
So, the simplified form is:
[tex]\[
\frac{x - 1}{x^2 + 3x + 2}
\][/tex]
### Expression 2:
[tex]\[
\frac{x - 1}{6x + 4}
\][/tex]
1. Factor the denominator:
[tex]\[
6x + 4 = 2(3x + 2)
\][/tex]
2. Rewrite the expression as:
[tex]\[
\frac{x - 1}{2(3x + 2)}
\][/tex]
So, the simplified form is:
[tex]\[
\frac{x - 1}{2(3x + 2)}
\][/tex]
### Expression 3:
[tex]\[
\frac{-1}{4x + 2}
\][/tex]
1. Factor the denominator:
[tex]\[
4x + 2 = 2(2x + 1)
\][/tex]
2. Rewrite the expression as:
[tex]\[
\frac{-1}{2(2x + 1)}
\][/tex]
So, the simplified form is:
[tex]\[
\frac{-1}{4x + 2}
\][/tex]
### Expression 4:
[tex]\[
\frac{1}{x + 2}
\][/tex]
There is no further simplification needed for this expression.
### Expression 5:
[tex]\[
\frac{x - 1}{x^2 + 3x + 2}
\][/tex]
Since \(x^2 + 3x + 2\) factors as \((x + 2)(x + 1)\), it is already in its simplest form:
[tex]\[
\frac{x - 1}{x^2 + 3x + 2}
\][/tex]
### Observation:
- Expression 1 simplifies to \(\frac{x - 1}{x^2 + 3x + 2}\).
- Expression 2 simplifies to \(\frac{x - 1}{2(3x + 2)}\).
- Expression 3 simplifies to \(\frac{-1}{4x + 2}\).
- Expression 4 is \(\frac{1}{x + 2}\).
- Expression 5 is \(\frac{x - 1}{x^2 + 3x + 2}\).
### Conclusion:
Comparing the simplified expressions, we observe that Expressions 1 and 5 are the same:
[tex]\[
\frac{x - 1}{x^2 + 3x + 2}
\][/tex]
Expression 2 is:
[tex]\[
\frac{x - 1}{2(3x + 2)}
\][/tex]
Expression 3 is:
[tex]\[
\frac{-1}{4x + 2}
\][/tex]
Expression 4 is:
[tex]\[
\frac{1}{x + 2}
\][/tex]
Thus, only Expressions 1 and 5 are identical; the others are distinct from each other.
