# Simplify (1/(2x^2)+x/4)/(1/(x+2)-x/2)

12×2+x41x+2-x2
Multiply the numerator and denominator of the complex fraction by 4(x+2)x2.
Multiply 12×2+x41x+2-x2 by 4(x+2)x24(x+2)x2.
4(x+2)x24(x+2)x2⋅12×2+x41x+2-x2
Combine.
4(x+2)x2(12×2+x4)4(x+2)x2(1x+2-x2)
4(x+2)x2(12×2+x4)4(x+2)x2(1x+2-x2)
Apply the distributive property.
4(x+2)x212x2+4(x+2)x2x44(x+2)x21x+2+4(x+2)x2(-x2)
Simplify by cancelling.
Cancel the common factor of 2×2.
Factor 2×2 out of 4(x+2)x2.
2×2(2(x+2))12×2+4(x+2)x2x44(x+2)x21x+2+4(x+2)x2(-x2)
Cancel the common factor.
2×2(2(x+2))12×2+4(x+2)x2x44(x+2)x21x+2+4(x+2)x2(-x2)
Rewrite the expression.
2(x+2)+4(x+2)x2x44(x+2)x21x+2+4(x+2)x2(-x2)
2(x+2)+4(x+2)x2x44(x+2)x21x+2+4(x+2)x2(-x2)
Cancel the common factor of 4.
Factor 4 out of 4(x+2)x2.
2(x+2)+4((x+2)x2)x44(x+2)x21x+2+4(x+2)x2(-x2)
Cancel the common factor.
2(x+2)+4((x+2)x2)x44(x+2)x21x+2+4(x+2)x2(-x2)
Rewrite the expression.
2(x+2)+(x+2)x2x4(x+2)x21x+2+4(x+2)x2(-x2)
2(x+2)+(x+2)x2x4(x+2)x21x+2+4(x+2)x2(-x2)
Raise x to the power of 1.
2(x+2)+(x+2)(x1x2)4(x+2)x21x+2+4(x+2)x2(-x2)
Use the power rule aman=am+n to combine exponents.
2(x+2)+(x+2)x1+24(x+2)x21x+2+4(x+2)x2(-x2)
2(x+2)+(x+2)x34(x+2)x21x+2+4(x+2)x2(-x2)
Cancel the common factor of x+2.
Factor x+2 out of 4(x+2)x2.
2(x+2)+(x+2)x3(x+2)(4×2)1x+2+4(x+2)x2(-x2)
Cancel the common factor.
2(x+2)+(x+2)x3(x+2)(4×2)1x+2+4(x+2)x2(-x2)
Rewrite the expression.
2(x+2)+(x+2)x34x2+4(x+2)x2(-x2)
2(x+2)+(x+2)x34x2+4(x+2)x2(-x2)
Cancel the common factor of 2.
Move the leading negative in -x2 into the numerator.
2(x+2)+(x+2)x34x2+4(x+2)x2-x2
Factor 2 out of 4(x+2)x2.
2(x+2)+(x+2)x34x2+2(2(x+2)x2)-x2
Cancel the common factor.
2(x+2)+(x+2)x34x2+2(2(x+2)x2)-x2
Rewrite the expression.
2(x+2)+(x+2)x34x2+2(x+2)x2(-x)
2(x+2)+(x+2)x34x2+2(x+2)x2(-x)
Multiply -1 by 2.
2(x+2)+(x+2)x34x2-2(x+2)x2x
Raise x to the power of 1.
2(x+2)+(x+2)x34x2-2(x+2)(x1x2)
Use the power rule aman=am+n to combine exponents.
2(x+2)+(x+2)x34x2-2(x+2)x1+2
2(x+2)+(x+2)x34x2-2(x+2)x3
2(x+2)+(x+2)x34x2-2(x+2)x3
Factor x+2 out of 2(x+2)+(x+2)x3.
Factor x+2 out of 2(x+2).
(x+2)⋅2+(x+2)x34x2-2(x+2)x3
Factor x+2 out of (x+2)x3.
(x+2)⋅2+(x+2)(x3)4×2-2(x+2)x3
Factor x+2 out of (x+2)⋅2+(x+2)(x3).
(x+2)(2+x3)4×2-2(x+2)x3
(x+2)(2+x3)4×2-2(x+2)x3
Simplify the denominator.
Factor 2×2 out of 4×2-2(x+2)x3.
Factor 2×2 out of 4×2.
(x+2)(2+x3)2×2(2)-2(x+2)x3
Factor 2×2 out of -2(x+2)x3.
(x+2)(2+x3)2×2(2)+2×2(-(x+2)x)
Factor 2×2 out of 2×2(2)+2×2(-(x+2)x).
(x+2)(2+x3)2×2(2-(x+2)x)
(x+2)(2+x3)2×2(2-(x+2)x)
Apply the distributive property.
(x+2)(2+x3)2×2(2+(-x-1⋅2)x)
Multiply -1 by 2.
(x+2)(2+x3)2×2(2+(-x-2)x)
Apply the distributive property.
(x+2)(2+x3)2×2(2-x⋅x-2x)
Multiply x by x by adding the exponents.
Move x.
(x+2)(2+x3)2×2(2-(x⋅x)-2x)
Multiply x by x.
(x+2)(2+x3)2×2(2-x2-2x)
(x+2)(2+x3)2×2(2-x2-2x)
Reorder terms.
(x+2)(2+x3)2×2(-x2-2x+2)
(x+2)(2+x3)2×2(-x2-2x+2)
Simplify with factoring out.
Factor -1 out of -x2.
(x+2)(2+x3)2×2(-(x2)-2x+2)
Factor -1 out of -2x.
(x+2)(2+x3)2×2(-(x2)-(2x)+2)
Factor -1 out of -(x2)-(2x).
(x+2)(2+x3)2×2(-(x2+2x)+2)
Rewrite 2 as -1(-2).
(x+2)(2+x3)2×2(-(x2+2x)-1(-2))
Factor -1 out of -(x2+2x)-1(-2).
(x+2)(2+x3)2×2(-(x2+2x-2))
Rewrite negatives.
Rewrite -(x2+2x-2) as -1(x2+2x-2).
(x+2)(2+x3)2×2(-1(x2+2x-2))
Move the negative in front of the fraction.
-(x+2)(2+x3)(2×2)(x2+2x-2)
Reorder factors in -(x+2)(2+x3)(2×2)(x2+2x-2).
-(x+2)(2+x3)2×2(x2+2x-2)
-(x+2)(2+x3)2×2(x2+2x-2)
-(x+2)(2+x3)2×2(x2+2x-2)
Simplify (1/(2x^2)+x/4)/(1/(x+2)-x/2)