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simplify the rational expression. State any restrictions on the variable. p^2 - 4p - 32/p + 4 –p + 8; p ≠ –4 p – 8; p ≠ –4 –p – 8; p ≠ –4 p + 8; p ≠ –4

Mathematics
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1. fully factor everything. 2. cancel as muich as you can out 3. make sure it's fully simplified 4. look at ALL denominators you have come across, set equal to 0, and thus find restrictions. (p^2 -4p -32) / (p+4) (p-8)(p+4) / (p+4) (p+4)s cancel each other out, leaving you with (p-8) / 1 p -8 Denominators: (p+4) p+4 =0 p= -4 THEREFORE p is every real number except for -4. (n^4 -11n^2 +30) / (n^4 -7n^2 +10) (n^2 -6)(n^2-5) / (n^2-5)(n^2 -2) NOTE that (n^2-5)s cancel each other out, leaving you with (n^2 -6) / (n^2 -2) can't simplify further. DENOMINATORS: (n^2-5)(n^2-2) =0 n^2-5 =0 and n^2 -2 = 0 n^2 = 5 and n^2 = 2 n= +/- √5 and n= +/- √2 THEREFORE n is all real numbers except for +/- √2, +/- √5 (x-4)(x+4) / (x+3)(x+2) / (x+4)(x+1) / (x-4)(x+2) WHICH IS THE SAME AS: (x-4)(x+4) / (x+3)(x+2) TIMES (x-4)(x+2) / (x+4)(x+1) (x+4)s cross cancel, (x+2)s cross cancel: (x-4) / (x+3) TIMES (x-4) / (x+1) (x-4)(x-4) / (x+3)(x+1) can't simplify further, denominators: (x+3)(x+2)(x+4)(x+1) =0 x+3=0 and x+2=0 and x+4=0 and x+1=0 x= -3 and x= -2 and x= -4 and x= -1 THEREFORE x is all real numbers except for -4, -3, -2, -1
\[\frac{ (p - 8)\cancel {(p + 4)} }{ \cancel {p + 4} }\]So, the only value you cannot have is: p = -4 Answer is "b"
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The slashes are for the cancellation. You are left with "p - 8"

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