If (4x+12)/((x+3)(x+4)) can be simplified to 4/(x+4), then wouldn't it only be undefined at x = -4 (and not x = -3)?

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At x = -3 you have a "hole".  Agree, it is not a vertical asymptote, but it is still not included in the x-domain and you have a "discontinuity" at x= - 3.Hope that helps