ANSWER
[tex]\begin{gathered} z\text{ = -}\frac{17}{20}\text{ + }\frac{19}{20}i \\ \text{option D} \end{gathered}[/tex]STEP-BY-STEP EXPLANATION:
Given information
[tex]z\text{ = }\frac{4\text{ + 7i}}{2\text{ - 6i}}[/tex]Step 1: We need to find the conjugate of the denominator
[tex]\begin{gathered} \text{The conjugate of 2 - 6i is 2 + 6i} \\ \text{Conjugate = 2 + 6i} \end{gathered}[/tex]Step 2: Multiply the conjugate by the numerator and the denominator
[tex]\frac{4\text{ + 7i}}{2\text{ - 6i}}\times\text{ }\frac{2\text{ +6i}}{2\text{ + 6i}}[/tex]Step 3: Simplify the above expression
[tex]\begin{gathered} z=\frac{(4\times2)\text{ + (4}\times6i)\text{ + (7i }\times2)\text{ + (7i}\times6i)}{(2\times2)\text{ + (2}\times6i)-(6i\times2)-(6i\times6i)} \\ \text{ Expand the parentheses} \\ z=\frac{8+24i+14i+42i^2}{4+12i-12i-36i^2} \\ \text{Recall, }i^2\text{ = -1} \\ \text{Substitute i}^2\text{ = -1 into the expression} \\ z=\frac{8\text{ + 38i + 42(-1)}}{4\text{ - }36(-1)} \\ \\ z=\frac{8\text{ + 38i - 42}}{4\text{ + 36}} \\ \\ z=\frac{8\text{ - 42 + 38i}}{40} \\ z=\frac{-34\text{ + 38i}}{40} \\ z=\text{ }\frac{-34}{40}\text{ + }\frac{38i}{40} \\ z\text{ = -}\frac{17}{20}\text{ + }\frac{19}{20}i \end{gathered}[/tex]