Reduce (¹/_{1 - 4i} - ²/_{1 + i})(^{3 - 4i}/_{5 + i}) to the standard form.

(¹/_{1 - 4i} - ²/_{1 + i}) = [^{(1 + i) - 2(1 - 4i)}/_{(1 - 4i)(1 + i)}] [^{3 - 4i}/_{5 + i}]

= [^{1 + i - 2 + 8i}/_{1 + i - 4i - 4i²} ] [^{3 - 4i}/_{5 + i}] = [^{-1 + 9i}/_{5 - 3i}] [^{3 - 4i}/_{5 + i}]

= ^{-3 + 4i + 27i - 36i2} /_{25 + 5i - 15i - 3i²} = ^{33 + 31i}/_{28 - 10i} = ^{33 + 31i}/_{2(14 - 5i)}

= ^{(33 + 31i)}/_{2(14 - 5i)} × ^{(14 + 5i)}/_{(14 + 5i)}   

[On multiplying numerator and denominator by (14 + 5i)]

= ^{462 + 165i + 434i + 155i²} /_{2[(14)² - (5i)²]} = ^{307 + 599i}/_{2(196 - 25i²)}

= ^{307 + 599i}/₂₍₂₂₁₎ = ^{307 + 599i}/₄₄₂ = ³⁰⁷/₄₄₂ + ⁵⁹⁹ⁱ/₄₄₂

This is the required standard form.