The perimeter of a triangle is `8y^(2) - 9y + 4` and its two sides are `3y^(2) - 5y` and `4y^(2) + 12`. Find its third side.

Answer

To find the third side of the triangle given the perimeter and two sides, we can follow these steps:

### Step 1: Write down the perimeter and the two sides
The perimeter of the triangle is given as:
\[ P = 8y^2 - 9y + 4 \]

The two sides of the triangle are:
\[ S_1 = 3y^2 - 5y \]
\[ S_2 = 4y^2 + 12 \]

### Step 2: Find the sum of the two sides
Now, we need to find the sum of the two sides:
\[ S_1 + S_2 = (3y^2 - 5y) + (4y^2 + 12) \]

Combine like terms:
\[ S_1 + S_2 = 3y^2 + 4y^2 - 5y + 12 = 7y^2 - 5y + 12 \]

### Step 3: Set up the equation for the third side
The perimeter of the triangle is equal to the sum of all three sides:
\[ P = S_1 + S_2 + S_3 \]
Where \( S_3 \) is the third side we need to find.

We can rearrange this to find \( S_3 \):
\[ S_3 = P - (S_1 + S_2) \]

### Step 4: Substitute the values
Substituting the values we have:
\[ S_3 = (8y^2 - 9y + 4) - (7y^2 - 5y + 12) \]

### Step 5: Simplify the expression
Now, we will simplify the expression for \( S_3 \):
\[ S_3 = 8y^2 - 9y + 4 - 7y^2 + 5y - 12 \]

Combine like terms:
\[ S_3 = (8y^2 - 7y^2) + (-9y + 5y) + (4 - 12) \]
\[ S_3 = 1y^2 - 4y - 8 \]

### Step 6: Write the final answer
Thus, the third side of the triangle is:
\[ S_3 = y^2 - 4y - 8 \]