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Simplify (5x-3)/(7x+3)+(8x^2+3)/(7x+1)

5x-37x+3+8x2+37x+1

Step 1
To write 5x-37x+3 as a fraction with a common denominator, multiply by 7x+17x+1.
5x-37x+37x+17x+1+8x2+37x+1

Step 2
To write 8x2+37x+1 as a fraction with a common denominator, multiply by 7x+37x+3.
5x-37x+37x+17x+1+8x2+37x+17x+37x+3

Step 3
Write each expression with a common denominator of (7x+3)(7x+1), by multiplying each by an appropriate factor of 1.

Multiply 5x-37x+3 by 7x+17x+1.
(5x-3)(7x+1)(7x+3)(7x+1)+8x2+37x+17x+37x+3
Multiply 8x2+37x+1 by 7x+37x+3.
(5x-3)(7x+1)(7x+3)(7x+1)+(8x2+3)(7x+3)(7x+1)(7x+3)
Reorder the factors of (7x+3)(7x+1).
(5x-3)(7x+1)(7x+1)(7x+3)+(8x2+3)(7x+3)(7x+1)(7x+3)
(5x-3)(7x+1)(7x+1)(7x+3)+(8x2+3)(7x+3)(7x+1)(7x+3)

Step 4
Combine the numerators over the common denominator.
(5x-3)(7x+1)+(8x2+3)(7x+3)(7x+1)(7x+3)

Step 5
Simplify the numerator.

Expand (5x-3)(7x+1) using the FOIL Method.

Apply the distributive property.
5x(7x+1)-3(7x+1)+(8x2+3)(7x+3)(7x+1)(7x+3)
Apply the distributive property.
5x(7x)+5x1-3(7x+1)+(8x2+3)(7x+3)(7x+1)(7x+3)
Apply the distributive property.
5x(7x)+5x1-3(7x)-31+(8x2+3)(7x+3)(7x+1)(7x+3)
5x(7x)+5x1-3(7x)-31+(8x2+3)(7x+3)(7x+1)(7x+3)
Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.
57xx+5x1-3(7x)-31+(8x2+3)(7x+3)(7x+1)(7x+3)
Multiply x by x by adding the exponents.

Move x.
57(xx)+5x1-3(7x)-31+(8x2+3)(7x+3)(7x+1)(7x+3)
Multiply x by x.
57x2+5x1-3(7x)-31+(8x2+3)(7x+3)(7x+1)(7x+3)
57x2+5x1-3(7x)-31+(8x2+3)(7x+3)(7x+1)(7x+3)
Multiply 5 by 7.
35x2+5x1-3(7x)-31+(8x2+3)(7x+3)(7x+1)(7x+3)
Multiply 5 by 1.
35x2+5x-3(7x)-31+(8x2+3)(7x+3)(7x+1)(7x+3)
Multiply 7 by -3.
35x2+5x-21x-31+(8x2+3)(7x+3)(7x+1)(7x+3)
Multiply -3 by 1.
35x2+5x-21x-3+(8x2+3)(7x+3)(7x+1)(7x+3)
35x2+5x-21x-3+(8x2+3)(7x+3)(7x+1)(7x+3)
Subtract 21x from 5x.
35x2-16x-3+(8x2+3)(7x+3)(7x+1)(7x+3)
35x2-16x-3+(8x2+3)(7x+3)(7x+1)(7x+3)
Expand (8x2+3)(7x+3) using the FOIL Method.

Apply the distributive property.
35x2-16x-3+8x2(7x+3)+3(7x+3)(7x+1)(7x+3)
Apply the distributive property.
35x2-16x-3+8x2(7x)+8x23+3(7x+3)(7x+1)(7x+3)
Apply the distributive property.
35x2-16x-3+8x2(7x)+8x23+3(7x)+33(7x+1)(7x+3)
35x2-16x-3+8x2(7x)+8x23+3(7x)+33(7x+1)(7x+3)
Simplify each term.

Rewrite using the commutative property of multiplication.
35x2-16x-3+87x2x+8x23+3(7x)+33(7x+1)(7x+3)
Multiply x2 by x by adding the exponents.

Move x.
35x2-16x-3+87(xx2)+8x23+3(7x)+33(7x+1)(7x+3)
Multiply x by x2.

Raise x to the power of 1.
35x2-16x-3+87(x1x2)+8x23+3(7x)+33(7x+1)(7x+3)
Use the power rule aman=am+n to combine exponents.
35x2-16x-3+87x1+2+8x23+3(7x)+33(7x+1)(7x+3)
35x2-16x-3+87x1+2+8x23+3(7x)+33(7x+1)(7x+3)
Add 1 and 2.
35x2-16x-3+87x3+8x23+3(7x)+33(7x+1)(7x+3)
35x2-16x-3+87x3+8x23+3(7x)+33(7x+1)(7x+3)
Multiply 8 by 7.
35x2-16x-3+56x3+8x23+3(7x)+33(7x+1)(7x+3)
Multiply 3 by 8.
35x2-16x-3+56x3+24x2+3(7x)+33(7x+1)(7x+3)
Multiply 7 by 3.
35x2-16x-3+56x3+24x2+21x+33(7x+1)(7x+3)
Multiply 3 by 3.
35x2-16x-3+56x3+24x2+21x+9(7x+1)(7x+3)
35x2-16x-3+56x3+24x2+21x+9(7x+1)(7x+3)
Add 35x2 and 24x2.
59x2-16x-3+56x3+21x+9(7x+1)(7x+3)
Add -16x and 21x.
59x2+5x-3+56x3+9(7x+1)(7x+3)
Add -3 and 9.
59x2+5x+56x3+6(7x+1)(7x+3)
Reorder terms.
56x3+59x2+5x+6(7x+1)(7x+3)
56x3+59x2+5x+6(7x+1)(7x+3)