# Simplify (10a+8-3a^2)/(a^2-a-12)*(9a^3-81a)/(3a^2-7a-6)

10a+8-3a2a2-a-12⋅9a3-81a3a2-7a-6
Factor by grouping.
Reorder terms.
-3a2+10a+8a2-a-12⋅9a3-81a3a2-7a-6
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=-3⋅8=-24 and whose sum is b=10.
Factor 10 out of 10a.
-3a2+10(a)+8a2-a-12⋅9a3-81a3a2-7a-6
Rewrite 10 as -2 plus 12
-3a2+(-2+12)a+8a2-a-12⋅9a3-81a3a2-7a-6
Apply the distributive property.
-3a2-2a+12a+8a2-a-12⋅9a3-81a3a2-7a-6
-3a2-2a+12a+8a2-a-12⋅9a3-81a3a2-7a-6
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(-3a2-2a)+12a+8a2-a-12⋅9a3-81a3a2-7a-6
Factor out the greatest common factor (GCF) from each group.
a(-3a-2)-4(-3a-2)a2-a-12⋅9a3-81a3a2-7a-6
a(-3a-2)-4(-3a-2)a2-a-12⋅9a3-81a3a2-7a-6
Factor the polynomial by factoring out the greatest common factor, -3a-2.
(-3a-2)(a-4)a2-a-12⋅9a3-81a3a2-7a-6
(-3a-2)(a-4)a2-a-12⋅9a3-81a3a2-7a-6
Factor a2-a-12 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -12 and whose sum is -1.
-4,3
Write the factored form using these integers.
(-3a-2)(a-4)(a-4)(a+3)⋅9a3-81a3a2-7a-6
(-3a-2)(a-4)(a-4)(a+3)⋅9a3-81a3a2-7a-6
Simplify the numerator.
Factor 9a out of 9a3-81a.
Factor 9a out of 9a3.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a2)-81a3a2-7a-6
Factor 9a out of -81a.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a2)+9a(-9)3a2-7a-6
Factor 9a out of 9a(a2)+9a(-9).
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a2-9)3a2-7a-6
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a2-9)3a2-7a-6
Rewrite 9 as 32.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a2-32)3a2-7a-6
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=a and b=3.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)3a2-7a-6
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)3a2-7a-6
Factor by grouping.
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=3⋅-6=-18 and whose sum is b=-7.
Factor -7 out of -7a.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)3a2-7(a)-6
Rewrite -7 as 2 plus -9
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)3a2+(2-9)a-6
Apply the distributive property.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)3a2+2a-9a-6
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)3a2+2a-9a-6
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)(3a2+2a)-9a-6
Factor out the greatest common factor (GCF) from each group.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)a(3a+2)-3(3a+2)
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)a(3a+2)-3(3a+2)
Factor the polynomial by factoring out the greatest common factor, 3a+2.
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)(3a+2)(a-3)
(-3a-2)(a-4)(a-4)(a+3)⋅9a(a+3)(a-3)(3a+2)(a-3)
Cancel the common factor of a+3.
Factor a+3 out of (a-4)(a+3).
(-3a-2)(a-4)(a+3)(a-4)⋅9a(a+3)(a-3)(3a+2)(a-3)
Factor a+3 out of 9a(a+3)(a-3).
(-3a-2)(a-4)(a+3)(a-4)⋅(a+3)(9a(a-3))(3a+2)(a-3)
Cancel the common factor.
(-3a-2)(a-4)(a+3)(a-4)⋅(a+3)(9a(a-3))(3a+2)(a-3)
Rewrite the expression.
(-3a-2)(a-4)a-4⋅9a(a-3)(3a+2)(a-3)
(-3a-2)(a-4)a-4⋅9a(a-3)(3a+2)(a-3)
Multiply (-3a-2)(a-4)a-4 and 9a(a-3)(3a+2)(a-3).
(-3a-2)(a-4)(9a(a-3))(a-4)((3a+2)(a-3))
Cancel the common factor of -3a-2 and 3a+2.
Factor -1 out of -3a.
(-(3a)-2)(a-4)(9a(a-3))(a-4)((3a+2)(a-3))
Rewrite -2 as -1(2).
(-(3a)-1(2))(a-4)(9a(a-3))(a-4)((3a+2)(a-3))
Factor -1 out of -(3a)-1(2).
-(3a+2)(a-4)(9a(a-3))(a-4)((3a+2)(a-3))
Rewrite -(3a+2) as -1(3a+2).
-1(3a+2)(a-4)(9a(a-3))(a-4)((3a+2)(a-3))
Cancel the common factor.
-1(3a+2)(a-4)⋅9a(a-3)(a-4)(3a+2)(a-3)
Rewrite the expression.
(-1(a-4))⋅9a(a-3)(a-4)(a-3)
(-1(a-4))⋅9a(a-3)(a-4)(a-3)
Cancel the common factor of a-4.
Cancel the common factor.
-1(a-4)⋅9a(a-3)(a-4)(a-3)
Rewrite the expression.
(-1)⋅9a(a-3)a-3
(-1)⋅9a(a-3)a-3
Cancel the common factor of a-3.
Cancel the common factor.
(-1)⋅9a(a-3)a-3
Divide (-1)⋅9a by 1.
(-1)⋅9a
(-1)⋅9a
Multiply -1 by 9.
-9a
Simplify (10a+8-3a^2)/(a^2-a-12)*(9a^3-81a)/(3a^2-7a-6)

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