Add (x+1)/(x^2+6x+9)+(x-4)/(x^2-9)

Math
x+1×2+6x+9+x-4×2-9
Simplify each term.
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Factor using the perfect square rule.
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Rewrite 9 as 32.
x+1×2+6x+32+x-4×2-9
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
6x=2⋅x⋅3
Rewrite the polynomial.
x+1×2+2⋅x⋅3+32+x-4×2-9
Factor using the perfect square trinomial rule a2+2ab+b2=(a+b)2, where a=x and b=3.
x+1(x+3)2+x-4×2-9
x+1(x+3)2+x-4×2-9
Simplify the denominator.
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Rewrite 9 as 32.
x+1(x+3)2+x-4×2-32
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=x and b=3.
x+1(x+3)2+x-4(x+3)(x-3)
x+1(x+3)2+x-4(x+3)(x-3)
x+1(x+3)2+x-4(x+3)(x-3)
To write x+1(x+3)2 as a fraction with a common denominator, multiply by x-3x-3.
x+1(x+3)2⋅x-3x-3+x-4(x+3)(x-3)
To write x-4(x+3)(x-3) as a fraction with a common denominator, multiply by x+3x+3.
x+1(x+3)2⋅x-3x-3+x-4(x+3)(x-3)⋅x+3x+3
Write each expression with a common denominator of (x-3)(x+3)2, by multiplying each by an appropriate factor of 1.
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Multiply x+1(x+3)2 and x-3x-3.
(x+1)(x-3)(x+3)2(x-3)+x-4(x+3)(x-3)⋅x+3x+3
Multiply x-4(x+3)(x-3) and x+3x+3.
(x+1)(x-3)(x+3)2(x-3)+(x-4)(x+3)(x+3)(x-3)(x+3)
Raise x+3 to the power of 1.
(x+1)(x-3)(x+3)2(x-3)+(x-4)(x+3)(x+3)1(x+3)(x-3)
Raise x+3 to the power of 1.
(x+1)(x-3)(x+3)2(x-3)+(x-4)(x+3)(x+3)1(x+3)1(x-3)
Use the power rule aman=am+n to combine exponents.
(x+1)(x-3)(x+3)2(x-3)+(x-4)(x+3)(x+3)1+1(x-3)
Add 1 and 1.
(x+1)(x-3)(x+3)2(x-3)+(x-4)(x+3)(x+3)2(x-3)
(x+1)(x-3)(x+3)2(x-3)+(x-4)(x+3)(x+3)2(x-3)
Combine the numerators over the common denominator.
(x+1)(x-3)+(x-4)(x+3)(x+3)2(x-3)
Simplify the numerator.
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Expand (x+1)(x-3) using the FOIL Method.
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Apply the distributive property.
x(x-3)+1(x-3)+(x-4)(x+3)(x+3)2(x-3)
Apply the distributive property.
x⋅x+x⋅-3+1(x-3)+(x-4)(x+3)(x+3)2(x-3)
Apply the distributive property.
x⋅x+x⋅-3+1x+1⋅-3+(x-4)(x+3)(x+3)2(x-3)
x⋅x+x⋅-3+1x+1⋅-3+(x-4)(x+3)(x+3)2(x-3)
Simplify and combine like terms.
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Simplify each term.
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Multiply x by x.
x2+x⋅-3+1x+1⋅-3+(x-4)(x+3)(x+3)2(x-3)
Move -3 to the left of x.
x2-3⋅x+1x+1⋅-3+(x-4)(x+3)(x+3)2(x-3)
Multiply x by 1.
x2-3x+x+1⋅-3+(x-4)(x+3)(x+3)2(x-3)
Multiply -3 by 1.
x2-3x+x-3+(x-4)(x+3)(x+3)2(x-3)
x2-3x+x-3+(x-4)(x+3)(x+3)2(x-3)
Add -3x and x.
x2-2x-3+(x-4)(x+3)(x+3)2(x-3)
x2-2x-3+(x-4)(x+3)(x+3)2(x-3)
Expand (x-4)(x+3) using the FOIL Method.
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Apply the distributive property.
x2-2x-3+x(x+3)-4(x+3)(x+3)2(x-3)
Apply the distributive property.
x2-2x-3+x⋅x+x⋅3-4(x+3)(x+3)2(x-3)
Apply the distributive property.
x2-2x-3+x⋅x+x⋅3-4x-4⋅3(x+3)2(x-3)
x2-2x-3+x⋅x+x⋅3-4x-4⋅3(x+3)2(x-3)
Simplify and combine like terms.
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Simplify each term.
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Multiply x by x.
x2-2x-3+x2+x⋅3-4x-4⋅3(x+3)2(x-3)
Move 3 to the left of x.
x2-2x-3+x2+3⋅x-4x-4⋅3(x+3)2(x-3)
Multiply -4 by 3.
x2-2x-3+x2+3x-4x-12(x+3)2(x-3)
x2-2x-3+x2+3x-4x-12(x+3)2(x-3)
Subtract 4x from 3x.
x2-2x-3+x2-x-12(x+3)2(x-3)
x2-2x-3+x2-x-12(x+3)2(x-3)
Add x2 and x2.
2×2-2x-3-x-12(x+3)2(x-3)
Subtract x from -2x.
2×2-3x-3-12(x+3)2(x-3)
Subtract 12 from -3.
2×2-3x-15(x+3)2(x-3)
2×2-3x-15(x+3)2(x-3)
Add (x+1)/(x^2+6x+9)+(x-4)/(x^2-9)

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