Factor out the common term \(3\).
\[\frac{2x(x+4)}{{(x+1)}^{2}}-\frac{3(x-1)}{{(x+1)}^{2}}\]
Join the denominators.
\[\frac{2x(x+4)-3(x-1)}{{(x+1)}^{2}}\]
Expand.
\[\frac{2{x}^{2}+8x-3x+3}{{(x+1)}^{2}}\]
Collect like terms.
\[\frac{2{x}^{2}+(8x-3x)+3}{{(x+1)}^{2}}\]
Simplify \(2{x}^{2}+(8x-3x)+3\) to \(2{x}^{2}+5x+3\).
\[\frac{2{x}^{2}+5x+3}{{(x+1)}^{2}}\]
Split the second term in \(2{x}^{2}+5x+3\) into two terms.
Multiply the coefficient of the first term by the constant term.
\[2\times 3=6\]
Ask: Which two numbers add up to \(5\) and multiply to \(6\)?
Split \(5x\) as the sum of \(3x\) and \(2x\).
\[2{x}^{2}+3x+2x+3\]
\[\frac{2{x}^{2}+3x+2x+3}{{(x+1)}^{2}}\]
Factor out common terms in the first two terms, then in the last two terms.
\[\frac{x(2x+3)+(2x+3)}{{(x+1)}^{2}}\]
Factor out the common term \(2x+3\).
\[\frac{(2x+3)(x+1)}{{(x+1)}^{2}}\]
Simplify.
\[\frac{2x+3}{x+1}\]
(2*x+3)/(x+1)
