`y ^(2) + 2y - 3 =? `

Options

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

Answer

To factor the expression \( y^2 + 2y - 3 \), we can follow these steps:

### Step 1: Identify the coefficients
The given expression is \( y^2 + 2y - 3 \). Here, the coefficient of \( y^2 \) (the leading coefficient) is 1, the coefficient of \( y \) is 2, and the constant term is -3.

### Step 2: Multiply the leading coefficient and the constant term
We multiply the leading coefficient (1) by the constant term (-3):
\[ 1 \times (-3) = -3 \]

### Step 3: Find two numbers that multiply to -3 and add to 2
We need to find two numbers that multiply to -3 (the result from Step 2) and add up to 2 (the coefficient of \( y \)). The numbers that satisfy this condition are 3 and -1, since:
\[ 3 \times (-1) = -3 \quad \text{and} \quad 3 + (-1) = 2 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the expression \( y^2 + 2y - 3 \) by splitting the middle term (2y) into two terms using the numbers found in Step 3:
\[ y^2 + 3y - 1y - 3 \]

### Step 5: Factor by grouping
Now we will group the terms:
\[ (y^2 + 3y) + (-1y - 3) \]
Next, we factor out the common factors in each group:
\[ y(y + 3) - 1(y + 3) \]

### Step 6: Factor out the common binomial
Now we can see that \( (y + 3) \) is a common factor:
\[ (y + 3)(y - 1) \]

### Final Answer
Thus, the factorization of \( y^2 + 2y - 3 \) is:
\[ (y + 3)(y - 1) \]

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