If `a^(2) + b^(2)= 34 and ab= 12`, find:
`3(a +b)^(2) + 5(a-b)^(2)`

Answer

To solve the problem, we need to find the value of \(3(a + b)^2 + 5(a - b)^2\) given that \(a^2 + b^2 = 34\) and \(ab = 12\).

### Step-by-step Solution:

1. **Expand the expressions**:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
\[ (a - b)^2 = a^2 - 2ab + b^2 \]

2. **Substitute the values into the expansions**:
From the problem, we know:
- \(a^2 + b^2 = 34\)
- \(ab = 12\)

Therefore:
\[ (a + b)^2 = a^2 + b^2 + 2ab = 34 + 2 \times 12 = 34 + 24 = 58 \]
\[ (a - b)^2 = a^2 + b^2 - 2ab = 34 - 2 \times 12 = 34 - 24 = 10 \]

3. **Substitute back into the original expression**:
Now we can substitute these values into the expression we want to evaluate:
\[ 3(a + b)^2 + 5(a - b)^2 = 3 \times 58 + 5 \times 10 \]

4. **Calculate**:
\[ 3 \times 58 = 174 \]
\[ 5 \times 10 = 50 \]
\[ 3(a + b)^2 + 5(a - b)^2 = 174 + 50 = 224 \]

### Final Answer:
\[ 3(a + b)^2 + 5(a - b)^2 = 224 \]