Question: Express as a complex number in simplest a+bi form:
$\frac{4 - 10i}{-3 - 7i}$
1. **State the problem:** Simplify the complex fraction $$\frac{4 - 10i}{-3 - 7i}$$ into the form $$a + bi$$ where $$a$$ and $$b$$ are real numbers.
2. **Formula and rule:** To simplify a complex fraction, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$-3 - 7i$$ is $$-3 + 7i$$.
3. **Multiply numerator and denominator by the conjugate:**
$$\frac{4 - 10i}{-3 - 7i} \times \frac{-3 + 7i}{-3 + 7i} = \frac{(4 - 10i)(-3 + 7i)}{(-3 - 7i)(-3 + 7i)}$$
4. **Calculate the denominator:**
$$(-3 - 7i)(-3 + 7i) = (-3)^2 - (7i)^2 = 9 - (49i^2) = 9 - 49(-1) = 9 + 49 = 58$$
5. **Calculate the numerator:**
$$ (4)(-3) + (4)(7i) + (-10i)(-3) + (-10i)(7i) = -12 + 28i + 30i - 70i^2 $$
Simplify the terms:
$$ -12 + (28i + 30i) - 70(-1) = -12 + 58i + 70 = ( -12 + 70 ) + 58i = 58 + 58i $$
6. **Write the fraction:**
$$ \frac{58 + 58i}{58} $$
7. **Simplify by dividing numerator and denominator by 58:**
$$ \frac{\cancel{58} + \cancel{58}i}{\cancel{58}} = 1 + i $$
**Final answer:** $$1 + i$$