(2+i)x+(1+2i)=1-4i

To solve the equation \((2+i)x + (1+2i) = 1-4i\), we will first isolate \(x\).

1. Start by moving \((1 + 2i)\) to the right side of the equation:

\[ (2+i)x = (1-4i) - (1+2i) \]

2. Simplify the right side:

\[ (1 - 4i) - (1 + 2i) = 1 - 4i - 1 - 2i = -6i \]

So the equation now is:

\[ (2+i)x = -6i \]

3. Now, solve for \(x\) by dividing both sides by \((2+i)\):

\[ x = \frac{-6i}{2+i} \]

4. To divide by a complex number, multiply the numerator and the denominator by the conjugate of the denominator:

\[ x = \frac{-6i(2-i)}{(2+i)(2-i)} \]

5. Calculate the denominator:

\[ (2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

6. Now calculate the numerator:

\[ -6i(2-i) = -12i + 6 = 6 - 12i \]

7. Putting it all together:

\[ x = \frac{6 - 12i}{5} = \frac{6}{5} - \frac{12}{5}i \]

Thus, the solution is:

\[ x = \frac{6}{5} - \frac{12}{5}i \]