Final answer:
To write the complex numbers in the form a + bi, we need to perform multiplication and division according to the given expressions. For the expression (2 - 3i)(4 + i), after expanding and simplifying, the complex number in the form a + bi is 5 - 14i. For the expression [(5 - 4i) - (3 + 7i)] / [(4 + 2i) + (2 - 3i)], after expanding, combining like terms, and rationalizing the denominator, the complex number in the form a + bi is (23 - 21i) / 37.
Step-by-step explanation:
To write the complex number in the form a + bi, we can perform the multiplication and division as follows:
a) (2 - 3i)(4 + i)
Expanding the expression, we get: 8 - 2i - 12i + 3i²
Combining like terms and using the fact that i² = -1, we get: (8 - 3) + (-2i - 12i)
Simplifying further, we have: 5 - 14i
Therefore, the complex number in the form a + bi is 5 - 14i.
b) [(5 - 4i) - (3 + 7i)] / [(4 + 2i) + (2 - 3i)]
Expanding the numerator and denominator first, we get: (5 - 4i - 3 - 7i) / (4 + 2i + 2 - 3i)
Combining like terms in both the numerator and denominator, we have: (2 - 11i) / (6 - i)
Next, we rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is 6 + i.
Multiplying, we get: (12 - 22i + i - 11i²) / (36 - i²)
Using the fact that i² = -1, we simplify further to: (12 - 21i - 11(-1)) / (36 - (-1))
Finally, combining like terms, the complex number in the form a + bi is: (23 - 21i) / 37.