If `(2x + 3y + 4) (2x + 3y -5)` is equivalent to `(ax^2 + by^2 + 2hxy + 2gx + 2fy + c)`, then what is the value of `(g+f-c) / (abh)`

Options

A. `35/432`
B. `19/108`
C. `19/216`
D. `37/216`

Answer

To solve the problem, we need to multiply the two expressions and then compare the resulting expression with the given equivalent expression. Let's go through the steps systematically.

### Step 1: Multiply the Expressions
We start with the expression:
\[ (2x + 3y + 4)(2x + 3y - 5) \]

Using the distributive property (also known as the FOIL method for binomials), we multiply the two expressions:

1. Multiply \(2x\) by each term in the second expression:
\[ 2x \cdot 2x = 4x^2 \]
\[ 2x \cdot 3y = 6xy \]
\[ 2x \cdot (-5) = -10x \]

2. Multiply \(3y\) by each term in the second expression:
\[ 3y \cdot 2x = 6xy \]
\[ 3y \cdot 3y = 9y^2 \]
\[ 3y \cdot (-5) = -15y \]

3. Multiply \(4\) by each term in the second expression:
\[ 4 \cdot 2x = 8x \]
\[ 4 \cdot 3y = 12y \]
\[ 4 \cdot (-5) = -20 \]

Now, we combine all these results:
\[ 4x^2 + 6xy + 6xy - 10x + 8x + 9y^2 - 15y + 12y - 20 \]

### Step 2: Combine Like Terms
Now we combine the like terms:
- For \(x^2\): \(4x^2\)
- For \(y^2\): \(9y^2\)
- For \(xy\): \(6xy + 6xy = 12xy\)
- For \(x\): \(-10x + 8x = -2x\)
- For \(y\): \(-15y + 12y = -3y\)
- Constant term: \(-20\)

Putting it all together, we have:
\[ 4x^2 + 9y^2 + 12xy - 2x - 3y - 20 \]

### Step 3: Compare with the Given Expression
The given expression is:
\[ ax^2 + by^2 + 2hxy + 2gx + 2fy + c \]
We need to match coefficients:
- \(a = 4\)
- \(b = 9\)
- \(2h = 12 \Rightarrow h = 6\)
- \(2g = -2 \Rightarrow g = -1\)
- \(2f = -3 \Rightarrow f = -\frac{3}{2}\)
- \(c = -20\)

### Step 4: Calculate \(g + f - c\)
Now we calculate:
\[ g + f - c = -1 - \frac{3}{2} - (-20) \]
\[ = -1 - \frac{3}{2} + 20 \]
Converting \(-1\) to a fraction:
\[ = -\frac{2}{2} - \frac{3}{2} + 20 = -\frac{5}{2} + 20 = -\frac{5}{2} + \frac{40}{2} = \frac{35}{2} \]

### Step 5: Calculate \(abh\)
Next, we calculate \(abh\):
\[ abh = 4 \times 9 \times 6 = 216 \]

### Step 6: Calculate \(\frac{g + f - c}{abh}\)
Finally, we compute:
\[ \frac{g + f - c}{abh} = \frac{\frac{35}{2}}{216} = \frac{35}{2 \times 216} = \frac{35}{432} \]

### Final Answer
Thus, the value of \(\frac{g + f - c}{abh}\) is:
\[ \frac{35}{432} \]