The product of two numbers is 12. If their sum added to the sum of their squares is 32, find the numbers.

Answer

To solve the problem, we need to find two numbers \( a \) and \( b \) that satisfy the following conditions:

1. The product of the two numbers is 12:
\[ ab = 12 \]

2. The sum of the two numbers added to the sum of their squares is 32:
\[ a + b + a^2 + b^2 = 32 \]

### Step 1: Express \( a^2 + b^2 \) in terms of \( a + b \) and \( ab \)

We know from algebra that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]

### Step 2: Substitute \( ab \) into the equation

From the first condition, we have \( ab = 12 \). Therefore, substituting this into the equation gives:
\[ a + b + (a + b)^2 - 2(12) = 32 \]
This simplifies to:
\[ a + b + (a + b)^2 - 24 = 32 \]

### Step 3: Rearranging the equation

Now, we can rearrange the equation:
\[ a + b + (a + b)^2 = 32 + 24 \]
\[ a + b + (a + b)^2 = 56 \]

### Step 4: Let \( x = a + b \)

Let \( x = a + b \). Then we can rewrite the equation as:
\[ x + x^2 = 56 \]
Rearranging gives:
\[ x^2 + x - 56 = 0 \]

### Step 5: Factor the quadratic equation

Next, we need to factor the quadratic equation \( x^2 + x - 56 = 0 \). We look for two numbers that multiply to \(-56\) and add to \(1\). The factors of \(-56\) that satisfy this are \(8\) and \(-7\):
\[ (x + 8)(x - 7) = 0 \]

### Step 6: Solve for \( x \)

Setting each factor to zero gives us:
\[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \quad \text{(not valid since } a + b \text{ must be positive)} \]
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]

### Step 7: Find \( a \) and \( b \)

Now that we have \( a + b = 7 \) and \( ab = 12 \), we can express \( b \) in terms of \( a \):
\[ b = 7 - a \]

Substituting this into the product equation:
\[ a(7 - a) = 12 \]
Expanding this gives:
\[ 7a - a^2 = 12 \]
Rearranging leads to:
\[ a^2 - 7a + 12 = 0 \]

### Step 8: Factor the quadratic equation

Now we need to factor \( a^2 - 7a + 12 = 0 \). The factors of \( 12 \) that add up to \( -7 \) are \( -3 \) and \( -4 \):
\[ (a - 3)(a - 4) = 0 \]

### Step 9: Solve for \( a \)

Setting each factor to zero gives us:
\[ a - 3 = 0 \quad \Rightarrow \quad a = 3 \]
\[ a - 4 = 0 \quad \Rightarrow \quad a = 4 \]

### Step 10: Find corresponding \( b \)

If \( a = 3 \), then:
\[ b = 7 - 3 = 4 \]
If \( a = 4 \), then:
\[ b = 7 - 4 = 3 \]

### Conclusion

Thus, the two numbers are \( 3 \) and \( 4 \).