Express the following in the form a+bi
`(sqrt3 + 5i) (sqrt3 -5i)^(2) + (-4+ 5i)^(2)`

Answer

To express the given expression \((\sqrt{3} + 5i)(\sqrt{3} - 5i)^2 + (-4 + 5i)^2\) in the form \(a + bi\), we will follow these steps:

### Step 1: Expand \((\sqrt{3} - 5i)^2\)
Using the formula for the square of a binomial, \((a - b)^2 = a^2 - 2ab + b^2\):
\[ (\sqrt{3} - 5i)^2 = (\sqrt{3})^2 - 2(\sqrt{3})(5i) + (5i)^2 \]
Calculating each term:
- \((\sqrt{3})^2 = 3\)
- \(-2(\sqrt{3})(5i) = -10\sqrt{3}i\)
- \((5i)^2 = 25i^2 = 25(-1) = -25\)

Combining these, we get:
\[ (\sqrt{3} - 5i)^2 = 3 - 10\sqrt{3}i - 25 = -22 - 10\sqrt{3}i \]

### Step 2: Substitute back into the expression
Now we substitute this back into the original expression:
\[ (\sqrt{3} + 5i)(-22 - 10\sqrt{3}i) + (-4 + 5i)^2 \]

### Step 3: Expand \((\sqrt{3} + 5i)(-22 - 10\sqrt{3}i)\)
Using the distributive property:
\[ (\sqrt{3} + 5i)(-22 - 10\sqrt{3}i) = \sqrt{3}(-22) + \sqrt{3}(-10\sqrt{3}i) + 5i(-22) + 5i(-10\sqrt{3}i) \]
Calculating each term:
- \(\sqrt{3}(-22) = -22\sqrt{3}\)
- \(\sqrt{3}(-10\sqrt{3}i) = -30i\)
- \(5i(-22) = -110i\)
- \(5i(-10\sqrt{3}i) = -50(-1) = 50\)

Combining these, we get:
\[ -22\sqrt{3} + 50 - (30 + 110)i = 50 - 22\sqrt{3} - 140i \]

### Step 4: Expand \((-4 + 5i)^2\)
Using the formula for the square of a binomial:
\[ (-4 + 5i)^2 = (-4)^2 + 2(-4)(5i) + (5i)^2 \]
Calculating each term:
- \((-4)^2 = 16\)
- \(2(-4)(5i) = -40i\)
- \((5i)^2 = 25i^2 = -25\)

Combining these, we have:
\[ (-4 + 5i)^2 = 16 - 40i - 25 = -9 - 40i \]

### Step 5: Combine all parts
Now we combine the results from Steps 3 and 4:
\[ (50 - 22\sqrt{3} - 140i) + (-9 - 40i) \]
Combining like terms:
- Real part: \(50 - 9 - 22\sqrt{3} = 41 - 22\sqrt{3}\)
- Imaginary part: \(-140 - 40 = -180\)

Thus, we have:
\[ 41 - 22\sqrt{3} - 180i \]

### Final Result
The expression in the form \(a + bi\) is:
\[ \boxed{(41 - 22\sqrt{3}) - 180i} \]