# Combine (t-10)/(t^2-t-6)-(t+5)/(t^2+3t+2)

t-10t2-t-6-t+5t2+3t+2
Simplify each term.
Factor t2-t-6 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 -6 and whose sum is -1.
-3,2
Write the factored form using these integers.
t-10(t-3)(t+2)-t+5t2+3t+2
t-10(t-3)(t+2)-t+5t2+3t+2
Factor t2+3t+2 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 2 and whose sum is 3.
1,2
Write the factored form using these integers.
t-10(t-3)(t+2)-t+5(t+1)(t+2)
t-10(t-3)(t+2)-t+5(t+1)(t+2)
t-10(t-3)(t+2)-t+5(t+1)(t+2)
To write t-10(t-3)(t+2) as a fraction with a common denominator, multiply by t+1t+1.
t-10(t-3)(t+2)⋅t+1t+1-t+5(t+1)(t+2)
To write -t+5(t+1)(t+2) as a fraction with a common denominator, multiply by t-3t-3.
t-10(t-3)(t+2)⋅t+1t+1-t+5(t+1)(t+2)⋅t-3t-3
Write each expression with a common denominator of (t-3)(t+2)(t+1), by multiplying each by an appropriate factor of 1.
Multiply t-10(t-3)(t+2) and t+1t+1.
(t-10)(t+1)(t-3)(t+2)(t+1)-t+5(t+1)(t+2)⋅t-3t-3
Multiply t+5(t+1)(t+2) and t-3t-3.
(t-10)(t+1)(t-3)(t+2)(t+1)-(t+5)(t-3)(t+1)(t+2)(t-3)
Reorder the factors of (t-3)(t+2)(t+1).
(t-10)(t+1)(t+1)(t+2)(t-3)-(t+5)(t-3)(t+1)(t+2)(t-3)
(t-10)(t+1)(t+1)(t+2)(t-3)-(t+5)(t-3)(t+1)(t+2)(t-3)
Combine the numerators over the common denominator.
(t-10)(t+1)-(t+5)(t-3)(t+1)(t+2)(t-3)
Simplify the numerator.
Expand (t-10)(t+1) using the FOIL Method.
Apply the distributive property.
t(t+1)-10(t+1)-(t+5)(t-3)(t+1)(t+2)(t-3)
Apply the distributive property.
t⋅t+t⋅1-10(t+1)-(t+5)(t-3)(t+1)(t+2)(t-3)
Apply the distributive property.
t⋅t+t⋅1-10t-10⋅1-(t+5)(t-3)(t+1)(t+2)(t-3)
t⋅t+t⋅1-10t-10⋅1-(t+5)(t-3)(t+1)(t+2)(t-3)
Simplify and combine like terms.
Simplify each term.
Multiply t by t.
t2+t⋅1-10t-10⋅1-(t+5)(t-3)(t+1)(t+2)(t-3)
Multiply t by 1.
t2+t-10t-10⋅1-(t+5)(t-3)(t+1)(t+2)(t-3)
Multiply -10 by 1.
t2+t-10t-10-(t+5)(t-3)(t+1)(t+2)(t-3)
t2+t-10t-10-(t+5)(t-3)(t+1)(t+2)(t-3)
Subtract 10t from t.
t2-9t-10-(t+5)(t-3)(t+1)(t+2)(t-3)
t2-9t-10-(t+5)(t-3)(t+1)(t+2)(t-3)
Apply the distributive property.
t2-9t-10+(-t-1⋅5)(t-3)(t+1)(t+2)(t-3)
Multiply -1 by 5.
t2-9t-10+(-t-5)(t-3)(t+1)(t+2)(t-3)
Expand (-t-5)(t-3) using the FOIL Method.
Apply the distributive property.
t2-9t-10-t(t-3)-5(t-3)(t+1)(t+2)(t-3)
Apply the distributive property.
t2-9t-10-t⋅t-t⋅-3-5(t-3)(t+1)(t+2)(t-3)
Apply the distributive property.
t2-9t-10-t⋅t-t⋅-3-5t-5⋅-3(t+1)(t+2)(t-3)
t2-9t-10-t⋅t-t⋅-3-5t-5⋅-3(t+1)(t+2)(t-3)
Simplify and combine like terms.
Simplify each term.
Multiply t by t by adding the exponents.
Move t.
t2-9t-10-(t⋅t)-t⋅-3-5t-5⋅-3(t+1)(t+2)(t-3)
Multiply t by t.
t2-9t-10-t2-t⋅-3-5t-5⋅-3(t+1)(t+2)(t-3)
t2-9t-10-t2-t⋅-3-5t-5⋅-3(t+1)(t+2)(t-3)
Multiply -3 by -1.
t2-9t-10-t2+3t-5t-5⋅-3(t+1)(t+2)(t-3)
Multiply -5 by -3.
t2-9t-10-t2+3t-5t+15(t+1)(t+2)(t-3)
t2-9t-10-t2+3t-5t+15(t+1)(t+2)(t-3)
Subtract 5t from 3t.
t2-9t-10-t2-2t+15(t+1)(t+2)(t-3)
t2-9t-10-t2-2t+15(t+1)(t+2)(t-3)
Subtract t2 from t2.
-9t-10+0-2t+15(t+1)(t+2)(t-3)
-9t-10-2t+15(t+1)(t+2)(t-3)
Subtract 2t from -9t.
-11t-10+15(t+1)(t+2)(t-3)
-11t+5(t+1)(t+2)(t-3)
-11t+5(t+1)(t+2)(t-3)
Simplify with factoring out.
Factor -1 out of -11t.
-(11t)+5(t+1)(t+2)(t-3)
Rewrite 5 as -1(-5).
-(11t)-1(-5)(t+1)(t+2)(t-3)
Factor -1 out of -(11t)-1(-5).
-(11t-5)(t+1)(t+2)(t-3)
Simplify the expression.
Rewrite -(11t-5) as -1(11t-5).
-1(11t-5)(t+1)(t+2)(t-3)
Move the negative in front of the fraction.
-11t-5(t+1)(t+2)(t-3)
-11t-5(t+1)(t+2)(t-3)
-11t-5(t+1)(t+2)(t-3)
Combine (t-10)/(t^2-t-6)-(t+5)/(t^2+3t+2)

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