Divide. Express your answer in standard form. \( \frac{-4}{2+3 i} \)

To divide \(\frac{-4}{2 + 3i}\) and express the result in standard form \(a + bi\), follow these steps: 1. **Multiply by the Conjugate:** Multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. \[ \frac{-4}{2 + 3i} \times \frac{2 - 3i}{2 - 3i} = \frac{-4(2 - 3i)}{(2 + 3i)(2 - 3i)} \] 2. **Simplify the Denominator:** Use the difference of squares formula: \[ (2 + 3i)(2 - 3i) = 2^2 - (3i)^2 = 4 - 9i^2 = 4 - 9(-1) = 4 + 9 = 13 \] 3. **Expand the Numerator:** Distribute \(-4\) across \(2 - 3i\): \[ -4(2 - 3i) = -8 + 12i \] 4. **Combine the Results:** \[ \frac{-8 + 12i}{13} = \frac{-8}{13} + \frac{12}{13}i \] Thus, the expression in standard form is: \[ \boxed{ -\dfrac{8}{13} + \dfrac{12}{13}\,i } \]