Just for fun, solution for the https://www.youtube.com/watch?v=10WnvBk9sZc

task.js

'use strict';

// 'ABAZDC', 'BACBAD'  => 'ABAD'
// 'AGGTAB', 'GXTXAYB' => 'GTAB'
// 'aaaa', 'aa'        => 'aa'
// '', '...'           => ''
// 'ABBA', 'ABCABA'    => 'ABBA'

function longestSubseq (a, b) {
  if (typeof a !== 'string') throw Error(`${a} is not a string`)
  if (typeof b !== 'string') throw Error(`${b} is not a string`)

  let s1 = Array.from(a)
  let s2 = Array.from(b)

  return s1
    .map((_, i) => extractSubseq(s1.slice(i), s2))
    .reduce((acc, v) => v.length >= acc.length ? v : acc, '')
}

function extractSubseq (a, b) {
  if (!Array.isArray(a)) throw Error(`${a} is not an Array`)
  if (!Array.isArray(b)) throw Error(`${b} is not an Array`)

  let result = []
  let s2 = b.slice()
  for (let i = 0; i < a.length; i += 1) {
    let index = s2.indexOf(a[i])
    if (index !== -1) {
      s2 = s2.slice(index + 1, s2.length)
      result.push(a[i])
      continue
    }
  }
  return result.join('')
}

function main () {
  let actual = null

  actual = longestSubseq('ABAZDC', 'BACBAD')
  if (actual !== 'ABAD') {
    throw Error(`Mistmatch. longestSubseq('ABAZDC', 'BACBAD'): ${actual}`)
  }

  actual = longestSubseq('AGGTAB', 'GXTXAYB')
  if (actual !== 'GTAB') {
    throw Error(`Mistmatch. longestSubseq('AGGTAB', 'GXTXAYB'): ${actual}`)
  }

  actual = longestSubseq('aaaa', 'aa')
  if (actual !== 'aa') {
    throw Error(`Mistmatch. longestSubseq('aaaa', 'aa'): ${actual}`)
  }

  actual = longestSubseq('aaaa', 'aa')
  if (actual !== 'aa') {
    throw Error(`Mistmatch. longestSubseq('aaaa', 'aa'): ${actual}`)
  }

  actual = longestSubseq('', '...')
  if (actual !== '') {
    throw Error(`Mistmatch. longestSubseq('', '...'): ${actual}`)
  }

  actual = longestSubseq('ABBA', 'ABCABA')
  if (actual !== 'ABBA') {
    throw Error(`Mistmatch. longestSubseq('ABBA', 'ABCABA'): ${actual}`)
  }
}

main()