This is a more complicated form of \(\displaystyle \frac{1}{2}+\frac{1}{3}= \frac{3}{6}+\frac{2}{6}=\frac{5}{6}\)

Find the least common denominator (LCD) and convert each fraction to the LCD, then add the numerators.  Simplify as needed.

\(\displaystyle \frac{x+2}{x-1}+\frac{x-5}{x+3}=\frac{x+2}{x-1}\cdot \frac{x+3}{x+3}+\frac{x-5}{x+3}\cdot \frac{x-1}{x-1}\)

which is equivalent to \(\displaystyle \frac{(x+2)\cdot (x+3)+(x-5)\cdot (x-1)}{(x+3)\cdot (x-1)}\)

Simplify to get \(\displaystyle \frac{2x^{2}-x+11}{x^{2}+2x-3}\)

Multiply by the reciprocal of the second expression:

\(\displaystyle \frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}\)

\(\displaystyle \frac{x^2+7x+10}{x+2}\cdot \frac{x^2-5x+6}{x^2+2x-15}\)

Factor the expressions:

\(\displaystyle \frac{(x+5)(x+2)}{x+2}\cdot \frac{(x-3)(x-2)}{(x-3)(x+5)}\)

Remove common terms:

\(\displaystyle \frac{\mathbf{(x+5)(x+2)}}{\mathbf{x+2}}\cdot \frac{\mathbf{(x-3)}(x-2)}{\mathbf{(x-3)(x+5)}}\)

\(\displaystyle x-2\)

Begin by multiplying the left term by \(\displaystyle \frac{3n-1}{3n-1}\):

\(\displaystyle 3n+1+\frac{1}{3n-1}\)

\(\displaystyle 3n+1 (\frac{3n-1}{3n-1})+\frac{1}{3n-1}\)

Simplify:

\(\displaystyle \frac{(3n+1)(3n-1)+1}{3n-1}\)

\(\displaystyle \frac{9n^2-1+1}{3n-1}\)

\(\displaystyle \frac{9n^2}{3n-1}\)

Begin by combining the terms in the denominator:

\(\displaystyle \frac{\frac{x+y}{x}}{\frac{1}{x}+\frac{1}{y}}\)

\(\displaystyle \frac{\frac{x+y}{x}}{\frac{x+y}{xy}}\)

Multiply by the reciprocal of the denominator:

\(\displaystyle \frac{x+y}{x} \cdot \frac{xy}{x+y}\)

Remove like terms:

\(\displaystyle \frac{\mathbf{x+y}}{\mathbf{x}} \cdot \frac{\boldsymbol{x}y}{\mathbf{x+y}}\)

\(\displaystyle y\)

Create a common denominator of \(\displaystyle (n+1)\) in both the numerator and denominator:

\(\displaystyle \frac{n+5+\frac{3}{n+1}}{n-1-\frac{3}{n+1}}\)

\(\displaystyle \frac{\frac{(n+5)(n+1)+3}{n+1}}{\frac{(n-1)(n+1)-3}{n+1}}\)

Multiply by the reciprocal of the denominator:

\(\displaystyle \frac{(n+5)(n+1)+3}{n+1}\cdot \frac{n+1}{(n-1)(n+1)-3}\)

Simplify:

\(\displaystyle \frac{n^2+6n+8}{n+1}\cdot \frac{n+1}{n^2-4}\)

\(\displaystyle \frac{(n+4)(n+2)}{n+1}\cdot \frac{n+1}{(n-2)(n+2)}\)

Remove common terms:

\(\displaystyle \frac{(n+4)\mathbf{(n+2)}}{\mathbf{n+1}}\cdot \frac{\mathbf{n+1}}{(n-2)(\mathbf{n+2})}\)

\(\displaystyle \frac{n+4}{n-2}\)

Factor the expression:

\(\displaystyle \frac{(a-b)(a^2+ab+b^2)(a+b)}{(b-a)(b+a)(a^2+ab+b^2)}\)

Remove like terms:

\(\displaystyle \frac{(a-b)(\mathbf{a^2+ab+b^2})(\mathbf{a+b})}{(b-a)(\mathbf{b+a})(\mathbf{a^2+ab+b^2})}\)

\(\displaystyle \frac{a-b}{b-a}=\frac{-1(b-a)}{b-a}=-1\)

Multiply by the inverse of the denominator:

\(\displaystyle \frac{x^2-11x+24}{x^2-18x+80}\div \frac{x^2-9x+20}{x^2-15x+50}\)

\(\displaystyle \frac{x^2-11x+24}{x^2-18x+80}\cdot \frac{x^2-15x+50}{x^2-9x+20}\)

Factor:

\(\displaystyle \frac{(x-8)(x-3)}{(x-8)(x-10)}\cdot \frac{(x-5)(x-10)}{(x-5)(x-4)}\)

Remove like terms:

\(\displaystyle \frac{(\mathbf{x-8})(x-3)}{(\mathbf{x-8})(\mathbf{x-10})}\cdot \frac{(\mathbf{x-5})(\mathbf{x-10})}{(\mathbf{x-5})(x-4)}\)

\(\displaystyle \frac{x-3}{x-4}\)