{
  "problem": "Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \\neq 0$. $a = \\dfrac{-5x - 20}{x^2 - 4x - 45} \\times \\dfrac{6x + 30}{-3x - 12} $",
  "hints": [
    "First factor the quadratic.",
    "$a = \\dfrac{-5x - 20}{(x + 5)(x - 9)} \\times \\dfrac{6x + 30}{-3x - 12} $",
    "Then factor out any other terms.",
    "$a = \\dfrac{-5(x + 4)}{(x + 5)(x - 9)} \\times \\dfrac{6(x + 5)}{-3(x + 4)} $",
    "Then multiply the two numerators and multiply the two denominators.",
    "$a = \\dfrac{ -5(x + 4) \\times 6(x + 5) } { (x + 5)(x - 9) \\times -3(x + 4) } $",
    "$a = \\dfrac{ -30(x + 4)(x + 5)}{ -3(x + 5)(x - 9)(x + 4)} $",
    "Notice that",
    "$(x + 4)$",
    "and",
    "$(x + 5)$",
    "appear",
    "in both the numerator and denominator so we can cancel them.",
    "$a = \\dfrac{ -30(x + 4)\\cancel{(x + 5)}}{ -3\\cancel{(x + 5)}(x - 9)(x + 4)} $",
    "We are dividing by",
    "$x + 5$",
    ", so",
    "$x + 5 \\neq 0$",
    "Therefore,",
    "$x \\neq -5$",
    "$a = \\dfrac{ -30\\cancel{(x + 4)}\\cancel{(x + 5)}}{ -3\\cancel{(x + 5)}(x - 9)\\cancel{(x + 4)}} $",
    "We are dividing by",
    "$x + 4$",
    ", so",
    "$x + 4 \\neq 0$",
    "Therefore,",
    "$x \\neq -4$",
    "$a = \\dfrac{-30}{-3(x - 9)} $",
    "$a = \\dfrac{10}{x - 9} ; \\space x \\neq -5 ; \\space x \\neq -4 $"
  ]
}
{'source': 'AMPS', 'file': 'amps/khan/430/1608222440.json'}