# Divide (4/(2-x)+5/(x-2))÷(4/x+2/(x-2))

(42-x+5x-2)÷(4x+2x-2)
Rewrite the division as a fraction.
42-x+5x-24x+2x-2
Multiply the numerator and denominator of the complex fraction by (2-x)(x-2)x.
Multiply 42-x+5x-24x+2x-2 by (2-x)(x-2)x(2-x)(x-2)x.
(2-x)(x-2)x(2-x)(x-2)x⋅42-x+5x-24x+2x-2
Combine.
(2-x)(x-2)x(42-x+5x-2)(2-x)(x-2)x(4x+2x-2)
(2-x)(x-2)x(42-x+5x-2)(2-x)(x-2)x(4x+2x-2)
Apply the distributive property.
(2-x)(x-2)x42-x+(2-x)(x-2)x5x-2(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
Simplify by cancelling.
Cancel the common factor of 2-x.
Factor 2-x out of (2-x)(x-2)x.
(2-x)((x-2)x)42-x+(2-x)(x-2)x5x-2(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
Cancel the common factor.
(2-x)((x-2)x)42-x+(2-x)(x-2)x5x-2(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
Rewrite the expression.
(x-2)x⋅4+(2-x)(x-2)x5x-2(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
(x-2)x⋅4+(2-x)(x-2)x5x-2(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
Cancel the common factor of x-2.
Factor x-2 out of (2-x)(x-2)x.
(x-2)x⋅4+(x-2)((2-x)x)5x-2(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
Cancel the common factor.
(x-2)x⋅4+(x-2)((2-x)x)5x-2(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
Rewrite the expression.
(x-2)x⋅4+(2-x)x⋅5(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
(x-2)x⋅4+(2-x)x⋅5(2-x)(x-2)x4x+(2-x)(x-2)x2x-2
Cancel the common factor of x.
Factor x out of (2-x)(x-2)x.
(x-2)x⋅4+(2-x)x⋅5x((2-x)(x-2))4x+(2-x)(x-2)x2x-2
Cancel the common factor.
(x-2)x⋅4+(2-x)x⋅5x((2-x)(x-2))4x+(2-x)(x-2)x2x-2
Rewrite the expression.
(x-2)x⋅4+(2-x)x⋅5(2-x)(x-2)⋅4+(2-x)(x-2)x2x-2
(x-2)x⋅4+(2-x)x⋅5(2-x)(x-2)⋅4+(2-x)(x-2)x2x-2
Rewrite 2 as -1(-2).
(x-2)x⋅4+(2-x)x⋅5(-1(-2)-x)(x-2)⋅4+(2-x)(x-2)x2x-2
Factor -1 out of -x.
(x-2)x⋅4+(2-x)x⋅5(-1(-2)-(x))(x-2)⋅4+(2-x)(x-2)x2x-2
Factor -1 out of -1(-2)-(x).
(x-2)x⋅4+(2-x)x⋅5-1(-2+x)(x-2)⋅4+(2-x)(x-2)x2x-2
Reorder terms.
(x-2)x⋅4+(2-x)x⋅5-1(x-2)(x-2)⋅4+(2-x)(x-2)x2x-2
Raise x-2 to the power of 1.
(x-2)x⋅4+(2-x)x⋅5-1((x-2)1(x-2))⋅4+(2-x)(x-2)x2x-2
Raise x-2 to the power of 1.
(x-2)x⋅4+(2-x)x⋅5-1((x-2)1(x-2)1)⋅4+(2-x)(x-2)x2x-2
Use the power rule aman=am+n to combine exponents.
(x-2)x⋅4+(2-x)x⋅5-1(x-2)1+1⋅4+(2-x)(x-2)x2x-2
(x-2)x⋅4+(2-x)x⋅5-1(x-2)2⋅4+(2-x)(x-2)x2x-2
Multiply 4 by -1.
(x-2)x⋅4+(2-x)x⋅5-4(x-2)2+(2-x)(x-2)x2x-2
Cancel the common factor of x-2.
Factor x-2 out of (2-x)(x-2)x.
(x-2)x⋅4+(2-x)x⋅5-4(x-2)2+(x-2)((2-x)x)2x-2
Cancel the common factor.
(x-2)x⋅4+(2-x)x⋅5-4(x-2)2+(x-2)((2-x)x)2x-2
Rewrite the expression.
(x-2)x⋅4+(2-x)x⋅5-4(x-2)2+(2-x)x⋅2
(x-2)x⋅4+(2-x)x⋅5-4(x-2)2+(2-x)x⋅2
(x-2)x⋅4+(2-x)x⋅5-4(x-2)2+(2-x)x⋅2
Simplify the numerator.
Factor x out of (x-2)x⋅4+(2-x)x⋅5.
Factor x out of (x-2)x⋅4.
x((x-2)⋅4)+(2-x)x⋅5-4(x-2)2+(2-x)x⋅2
Factor x out of (2-x)x⋅5.
x((x-2)⋅4)+x((2-x)⋅5)-4(x-2)2+(2-x)x⋅2
Factor x out of x((x-2)⋅4)+x((2-x)⋅5).
x((x-2)⋅4+(2-x)⋅5)-4(x-2)2+(2-x)x⋅2
x((x-2)⋅4+(2-x)⋅5)-4(x-2)2+(2-x)x⋅2
Apply the distributive property.
x(x⋅4-2⋅4+(2-x)⋅5)-4(x-2)2+(2-x)x⋅2
Move 4 to the left of x.
x(4⋅x-2⋅4+(2-x)⋅5)-4(x-2)2+(2-x)x⋅2
Multiply -2 by 4.
x(4⋅x-8+(2-x)⋅5)-4(x-2)2+(2-x)x⋅2
Apply the distributive property.
x(4x-8+2⋅5-x⋅5)-4(x-2)2+(2-x)x⋅2
Multiply 2 by 5.
x(4x-8+10-x⋅5)-4(x-2)2+(2-x)x⋅2
Multiply 5 by -1.
x(4x-8+10-5x)-4(x-2)2+(2-x)x⋅2
Subtract 5x from 4x.
x(-x-8+10)-4(x-2)2+(2-x)x⋅2
x(-x+2)-4(x-2)2+(2-x)x⋅2
x(-x+2)-4(x-2)2+(2-x)x⋅2
Simplify the denominator.
Factor 2 out of -4(x-2)2+(2-x)x⋅2.
Factor 2 out of -4(x-2)2.
x(-x+2)2(-2(x-2)2)+(2-x)x⋅2
Factor 2 out of (2-x)x⋅2.
x(-x+2)2(-2(x-2)2)+2⋅((2-x)x)
Factor 2 out of 2(-2(x-2)2)+2⋅((2-x)x).
x(-x+2)2(-2(x-2)2+(2-x)x)
x(-x+2)2(-2(x-2)2+(2-x)x)
Let u=x. Substitute u for all occurrences of x.
x(-x+2)2(-3u2+10u-8)
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⋅-8=24 and whose sum is b=10.
Factor 10 out of 10u.
x(-x+2)2(-3u2+10(u)-8)
Rewrite 10 as 4 plus 6
x(-x+2)2(-3u2+(4+6)u-8)
Apply the distributive property.
x(-x+2)2(-3u2+4u+6u-8)
x(-x+2)2(-3u2+4u+6u-8)
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
x(-x+2)2((-3u2+4u)+6u-8)
Factor out the greatest common factor (GCF) from each group.
x(-x+2)2(u(-3u+4)-2(-3u+4))
x(-x+2)2(u(-3u+4)-2(-3u+4))
Factor the polynomial by factoring out the greatest common factor, -3u+4.
x(-x+2)2((-3u+4)(u-2))
x(-x+2)2((-3u+4)(u-2))
Replace all occurrences of u with x.
x(-x+2)2(-3x+4)(x-2)
x(-x+2)2(-3x+4)(x-2)
Cancel the common factor of -x+2 and x-2.
Factor -1 out of -x.
x(-(x)+2)2(-3x+4)(x-2)
Rewrite 2 as -1(-2).
x(-(x)-1(-2))2(-3x+4)(x-2)
Factor -1 out of -(x)-1(-2).
x(-(x-2))2(-3x+4)(x-2)
Rewrite -(x-2) as -1(x-2).
x(-1(x-2))2(-3x+4)(x-2)
Cancel the common factor.
x⋅-1(x-2)2(-3x+4)(x-2)
Rewrite the expression.
x⋅-12(-3x+4)
x⋅-12(-3x+4)
Move -1 to the left of x.
-1⋅x2(-3x+4)
Move the negative in front of the fraction.
-x2(-3x+4)
Factor -1 out of -3x.
-x2(-(3x)+4)
Rewrite 4 as -1(-4).
-x2(-(3x)-1(-4))
Factor -1 out of -(3x)-1(-4).
-x2(-(3x-4))
Rewrite -(3x-4) as -1(3x-4).
-x2(-1(3x-4))
Move the negative in front of the fraction.
–x2(3x-4)
Multiply -1 by -1.
1×2(3x-4)
Multiply x2(3x-4) by 1.
x2(3x-4)
Divide (4/(2-x)+5/(x-2))÷(4/x+2/(x-2))