`(y^(3)+3y^(2)-y-3)/(y^(2)+4y+3)`
The expression above is equivalent to

Options

A. `y-1`
B. `y+1`
C. `(y-1)/(y+3)`
D. `y^(2)-1`

Answer

To solve the inequality \((y^{3}+3y^{2}-y-3)/(y^{2}+4y+3) < 0\), we will first factor both the numerator and the denominator.

### Step 1: Factor the numerator

The numerator is \(y^{3} + 3y^{2} - y - 3\). We can group the terms:

\[ y^{3} + 3y^{2} - y - 3 = (y^{3} + 3y^{2}) + (-y - 3) \]

Now, factor out common terms from each group:

\[ = y^{2}(y + 3) - 1(y + 3) \]

Now, we can factor out \((y + 3)\):

\[ = (y + 3)(y^{2} - 1) \]

Next, we can factor \(y^{2} - 1\) further using the difference of squares:

\[ y^{2} - 1 = (y - 1)(y + 1) \]

So, the complete factorization of the numerator is:

\[ y^{3} + 3y^{2} - y - 3 = (y + 3)(y - 1)(y + 1) \]

### Step 2: Factor the denominator

Now we factor the denominator \(y^{2} + 4y + 3\). We can look for two numbers that multiply to \(3\) and add up to \(4\):

\[ y^{2} + 4y + 3 = (y + 1)(y + 3) \]

### Step 3: Rewrite the expression

Now we can rewrite the original expression:

\[ \frac{(y + 3)(y - 1)(y + 1)}{(y + 1)(y + 3)} \]

### Step 4: Simplify the expression

We can cancel out the common factors \((y + 3)\) and \((y + 1)\) from the numerator and denominator:

\[ = y - 1 \]

### Step 5: Determine the inequality

Now we need to solve:

\[ y - 1 < 0 \]

This simplifies to:

\[ y < 1 \]

### Conclusion

The expression \((y^{3}+3y^{2}-y-3)/(y^{2}+4y+3)\) is equivalent to \(y - 1\), and the solution to the inequality is \(y < 1\).