Perform the indicated operation and give your answer in the form `x+yi`, where x and y are real numbers and `i= sqrt(-1)`
`(sqrt5-7i) (sqrt5-7i)^(2) + (-2+ 7i)^(2)`

Answer

To solve the expression \((\sqrt{5} - 7i)(\sqrt{5} - 7i)^{2} + (-2 + 7i)^{2}\), we will perform the operations step by step.

### Step 1: Simplify \((\sqrt{5} - 7i)^{2}\)

Using the formula \((a - b)^{2} = a^{2} - 2ab + b^{2}\):
- Let \(a = \sqrt{5}\) and \(b = 7i\).

\[ (\sqrt{5} - 7i)^{2} = (\sqrt{5})^{2} - 2(\sqrt{5})(7i) + (7i)^{2} \]

Calculating each term:
- \((\sqrt{5})^{2} = 5\)
- \(-2(\sqrt{5})(7i) = -14\sqrt{5}i\)
- \((7i)^{2} = 49i^{2} = 49(-1) = -49\)

Putting it all together:
\[ (\sqrt{5} - 7i)^{2} = 5 - 14\sqrt{5}i - 49 = -44 - 14\sqrt{5}i \]

### Step 2: Now compute \((\sqrt{5} - 7i)(-44 - 14\sqrt{5}i)\)

Using the distributive property:
\[ (\sqrt{5} - 7i)(-44 - 14\sqrt{5}i) = \sqrt{5}(-44) + \sqrt{5}(-14\sqrt{5}i) - 7i(-44) - 7i(-14\sqrt{5}i) \]

Calculating each term:
- \(\sqrt{5}(-44) = -44\sqrt{5}\)
- \(\sqrt{5}(-14\sqrt{5}i) = -14 \cdot 5i = -70i\)
- \(-7i(-44) = 308i\)
- \(-7i(-14\sqrt{5}i) = 98\)

Combining these results:
\[ -44\sqrt{5} + (308 - 70)i + 98 = -44\sqrt{5} + 238 + 308i \]

### Step 3: Compute \((-2 + 7i)^{2}\)

Using the formula \((a + b)^{2} = a^{2} + 2ab + b^{2}\):
- Let \(a = -2\) and \(b = 7i\).

\[ (-2 + 7i)^{2} = (-2)^{2} + 2(-2)(7i) + (7i)^{2} \]

Calculating each term:
- \((-2)^{2} = 4\)
- \(2(-2)(7i) = -28i\)
- \((7i)^{2} = 49i^{2} = -49\)

Putting it all together:
\[ (-2 + 7i)^{2} = 4 - 28i - 49 = -45 - 28i \]

### Step 4: Combine the results from Step 2 and Step 3

Now we add the two results:
\[ (-44\sqrt{5} + 238 + 308i) + (-45 - 28i) \]

Combining the real parts and the imaginary parts:
- Real part: \(-44\sqrt{5} + 238 - 45 = -44\sqrt{5} + 193\)
- Imaginary part: \(308i - 28i = 280i\)

### Final Result

Thus, the final answer in the form \(x + yi\) is:
\[ -44\sqrt{5} + 193 + 280i \]