Simplify the following complex numbers and find their modulus.
`=((2+4i)(1+2i))/((-1-i)(3-i))`

Answer

To simplify the complex number given by the expression \(\frac{(2 + 4i)(1 + 2i)}{(-1 - i)(3 - i)}\) and find its modulus, we can follow these steps:

### Step 1: Multiply the Numerator
We start by multiplying the two complex numbers in the numerator:

\[ (2 + 4i)(1 + 2i) = 2 \cdot 1 + 2 \cdot 2i + 4i \cdot 1 + 4i \cdot 2i \]

Calculating each term:

\[ = 2 + 4i + 4i + 8i^2 \]

Since \(i^2 = -1\), we have:

\[ = 2 + 8(-1) + 8i = 2 - 8 + 8i = -6 + 8i \]

### Step 2: Multiply the Denominator
Now, we multiply the two complex numbers in the denominator:

\[ (-1 - i)(3 - i) = -1 \cdot 3 + (-1)(-i) - i \cdot 3 - i(-i) \]

Calculating each term:

\[ = -3 + i - 3i + i^2 \]

Again, substituting \(i^2 = -1\):

\[ = -3 - 2i - 1 = -4 - 2i \]

### Step 3: Form the Complex Fraction
Now we can write our expression as:

\[ \frac{-6 + 8i}{-4 - 2i} \]

### Step 4: Rationalize the Denominator
To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{-6 + 8i}{-4 - 2i} \cdot \frac{-4 + 2i}{-4 + 2i} \]

Calculating the new numerator:

\[ (-6)(-4) + (-6)(2i) + (8i)(-4) + (8i)(2i) = 24 - 12i - 32i + 16i^2 \]

Substituting \(i^2 = -1\):

\[ = 24 - 44i + 16(-1) = 24 - 16 - 44i = 8 - 44i \]

Now calculating the new denominator:

\[ (-4)(-4) + (-4)(2i) + (-2i)(-4) + (-2i)(2i) = 16 - 8i + 8i - 4i^2 \]

Again substituting \(i^2 = -1\):

\[ = 16 + 4 = 20 \]

### Step 5: Final Expression
Now we have:

\[ \frac{8 - 44i}{20} \]

This simplifies to:

\[ \frac{8}{20} - \frac{44}{20}i = \frac{2}{5} - \frac{11}{5}i \]

### Step 6: Find the Modulus
The modulus of a complex number \(a + bi\) is given by:

\[ \sqrt{a^2 + b^2} \]

Here, \(a = \frac{2}{5}\) and \(b = -\frac{11}{5}\):

\[ \text{Modulus} = \sqrt{\left(\frac{2}{5}\right)^2 + \left(-\frac{11}{5}\right)^2} = \sqrt{\frac{4}{25} + \frac{121}{25}} = \sqrt{\frac{125}{25}} = \sqrt{5} \]

### Final Answer
The simplified form of the complex number is \(\frac{2}{5} - \frac{11}{5}i\) and its modulus is \(\sqrt{5}\).

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