` (5y +7)/((y +4) ^(2))-(5 )/((y +4))`
if the expression above the equal to `(-b)/((y+4) ^(2)),` where b is a positive constatn and `y ne -4,` what is the value of b ?

Options

A. 4
B. 7
C. 13
D. 27

Answer

To solve the problem step-by-step, we start with the given expression:

\[ \frac{5y + 7}{(y + 4)^2} - \frac{5}{(y + 4)} \]

### Step 1: Find a common denominator
The common denominator for the two fractions is \((y + 4)^2\). We rewrite the second fraction to have this common denominator:

\[ \frac{5y + 7}{(y + 4)^2} - \frac{5(y + 4)}{(y + 4)^2} \]

### Step 2: Simplify the expression
Now, we can combine the fractions:

\[ \frac{(5y + 7) - 5(y + 4)}{(y + 4)^2} \]

### Step 3: Distribute and combine like terms
Distributing the \(5\) in the second term gives us:

\[ \frac{(5y + 7) - (5y + 20)}{(y + 4)^2} \]

Now, combine the terms in the numerator:

\[ 5y + 7 - 5y - 20 = 7 - 20 = -13 \]

Thus, we have:

\[ \frac{-13}{(y + 4)^2} \]

### Step 4: Set the expression equal to the given form
According to the problem, this expression is equal to:

\[ \frac{-b}{(y + 4)^2} \]

### Step 5: Equate the numerators
From the equality of the two fractions, we can equate the numerators:

\[ -13 = -b \]

### Step 6: Solve for \(b\)
By multiplying both sides by \(-1\), we find:

\[ b = 13 \]

### Conclusion
Thus, the value of \(b\) is:

\[ \boxed{13} \]