Using properties of proportion find `x:y` given:
`(x^(2)+2x)/(2x+4)=(y^(2)+3y)/(3y+9)`

Answer

To solve the equation \(\frac{x^2 + 2x}{2x + 4} = \frac{y^2 + 3y}{3y + 9}\) and find the ratio \(x:y\), we will use the properties of proportions. Here’s the step-by-step solution:

### Step 1: Set up the equation
We start with the given proportion:
\[ \frac{x^2 + 2x}{2x + 4} = \frac{y^2 + 3y}{3y + 9} \]

### Step 2: Apply the property of componendo and dividendo
Using the property of componendo and dividendo, we can rewrite the equation as:
\[ \frac{(x^2 + 2x) + (2x + 4)}{(2x + 4) - (x^2 + 2x)} = \frac{(y^2 + 3y) + (3y + 9)}{(3y + 9) - (y^2 + 3y)} \]

### Step 3: Simplify the numerators and denominators
Calculating the left side:
- Numerator: \(x^2 + 2x + 2x + 4 = x^2 + 4x + 4\)
- Denominator: \(2x + 4 - (x^2 + 2x) = -x^2 + 2x + 4\)

Calculating the right side:
- Numerator: \(y^2 + 3y + 3y + 9 = y^2 + 6y + 9\)
- Denominator: \(3y + 9 - (y^2 + 3y) = -y^2 + 9\)

Thus, we have:
\[ \frac{x^2 + 4x + 4}{-x^2 + 2x + 4} = \frac{y^2 + 6y + 9}{-y^2 + 9} \]

### Step 4: Recognize the squares
Notice that:
- \(x^2 + 4x + 4 = (x + 2)^2\)
- \(-x^2 + 2x + 4 = -(x^2 - 2x - 4) = -(x - 2)^2\)

And similarly for \(y\):
- \(y^2 + 6y + 9 = (y + 3)^2\)
- \(-y^2 + 9 = -(y^2 - 9) = -(y - 3)(y + 3)\)

So we can rewrite the equation as:
\[ \frac{(x + 2)^2}{-(x - 2)(x + 2)} = \frac{(y + 3)^2}{-(y - 3)(y + 3)} \]

### Step 5: Cancel common terms
Canceling \((x + 2)\) and \((y + 3)\) from both sides gives us:
\[ \frac{x + 2}{-(x - 2)} = \frac{y + 3}{-(y - 3)} \]

### Step 6: Cross-multiply
Cross-multiplying gives:
\[ (x + 2)(y - 3) = (y + 3)(x - 2) \]

### Step 7: Expand and simplify
Expanding both sides:
\[ xy - 3x + 2y - 6 = xy - 2y + 3x - 6 \]

### Step 8: Rearranging terms
Rearranging gives:
\[ -3x + 2y = -2y + 3x \]
\[ 5x = 4y \]

### Step 9: Find the ratio
This simplifies to:
\[ \frac{x}{y} = \frac{4}{5} \]

Thus, the ratio \(x:y\) is:
\[ x:y = 4:5 \]

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