Divide `15 (y+ 3) ( y ^(2) - 16) by 6 ( y ^(2) - y -12).` s

Answer

To solve the problem of dividing \( 15(y + 3)(y^2 - 16) \) by \( 6(y^2 - y - 12) \), we will follow these steps:

### Step 1: Write the expression in a proper form
We start with the expression:
\[ \frac{15(y + 3)(y^2 - 16)}{6(y^2 - y - 12)} \]

### Step 2: Factor the numerator
In the numerator, we have \( y^2 - 16 \). This can be factored using the difference of squares:
\[ y^2 - 16 = (y - 4)(y + 4) \]
Thus, the numerator becomes:
\[ 15(y + 3)(y - 4)(y + 4) \]

### Step 3: Factor the denominator
In the denominator, we need to factor \( y^2 - y - 12 \). We will use the middle term splitting method. We look for two numbers that multiply to \(-12\) (the constant term) and add to \(-1\) (the coefficient of \(y\)). The numbers that satisfy this are \(3\) and \(-4\).

We can split the middle term:
\[ y^2 - y - 12 = y^2 - 4y + 3y - 12 \]
Now, we can group the terms:
\[ = y(y - 4) + 3(y - 4) \]
Factoring out the common term \((y - 4)\):
\[ = (y - 4)(y + 3) \]

### Step 4: Rewrite the expression with the factored forms
Now we can rewrite the original expression:
\[ \frac{15(y + 3)(y - 4)(y + 4)}{6(y - 4)(y + 3)} \]

### Step 5: Cancel common factors
We can cancel the common factors \((y + 3)\) and \((y - 4)\) from the numerator and the denominator:
\[ = \frac{15(y + 4)}{6} \]

### Step 6: Simplify the expression
Now we simplify the fraction:
\[ = \frac{15}{6}(y + 4) = \frac{5}{2}(y + 4) \]

### Final Answer
Thus, the final answer is:
\[ \frac{5}{2}(y + 4) \]

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