To simplify the expression, we need to find a common denominator for all the fractions.
StartFraction x + 5 Over x + 2 EndFraction minus StartFraction x + 1 Over x squared + 2 x EndFraction
= (x + 5)/(x + 2) - (x + 1)/(x(x + 2))
Next, we can combine the two fractions by finding a common denominator.
= [(x + 5)x - (x + 1)]/(x(x + 2))
= (x^2 + 4x - 1)/(x(x + 2))
StartFraction x squared + 4 x minus 1 Over x (x + 2) EndFraction StartFraction x squared + 4 x + 1 Over x (x + 2) EndFraction
We can combine these two fractions by adding the numerators and keeping the same denominator.
= (x^2 + 4x - 1)/(x(x + 2)) + (x^2 + 4x + 1)/(x(x + 2))
= (2x^2 + 8x)/(x(x + 2))
StartFraction 4 Over negative 1 (x squared + x minus 2) EndFraction
= -4/(x^2 + x - 2)
StartFraction x squared + 6 x + 1 Over x (x + 2) EndFraction
We can use partial fraction decomposition to split this fraction into simpler ones.
= (x + 3)/(x + 2) + (x + 1)/x
Now we can simplify each fraction separately.
= (x^2 + 5x + 6)/(x(x + 2)) + 1 + 1/x
= (x^2 + 5x + 6)/(x(x + 2)) + (x + 2)/(x(x + 2))
= (x^2 + 6x + 8)/(x(x + 2))
Now we can simplify the entire expression by combining all the fractions and finding a common denominator.
= (x^2 + 4x - 1)/(x(x + 2)) - 4/(x^2 + x - 2) + (x^2 + 6x + 8)/(x(x + 2))
= [((x^2 + 4x - 1) * (x^2 + x - 2)) - (4 * x(x + 2)) + ((x^2 + 6x + 8) * x)]/[x(x + 2)(x^2 + x - 2)]
= (x^4 + 6x^3 + 4x^2 - 7x - 8)/(x(x + 2)(x^2 + x - 2))
Therefore, the difference between the expressions is (x^4 + 6x^3 + 4x^2 - 7x - 8)/(x(x + 2)(x^2 + x - 2)).