\(\frac{2x}{x^2+4x+4}+\frac{x+1}{x+2}+\frac{2-x}{x^2+4x+4}\)

\(=\frac{2x}{\left(x+2\right)^2}+\frac{\left(x+1\right)\left(x+2\right)}{\left(x+2\right)^2}+\frac{2-x}{\left(x+2\right)^2}\)

\(=\frac{2x+x^2+3x+2+2-x}{\left(x+2\right)^2}\)

\(=\frac{x^2+4x+4}{\left(x+2\right)^2}\)

\(=\frac{\left(x+2\right)^2}{\left(x+2\right)^2}\)

\(=1\)

- Muốn cộng hai phân thức cùng mẫu, ta cộng các tử với nhau và giữ nguyên mẫu.

- Muốn cộng hai phân thức khác mẫu, ta quy đồng mẫu thức rồi cộng các phân thức cùng mẫu vừa tìm được.

\(\dfrac{3x}{x^3-1}+\dfrac{x-1}{x^2+x+1}\)

\(=\dfrac{3x}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{\left(x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(=\dfrac{3x+x^2-2x+1}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(=\dfrac{x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(=\dfrac{1}{x-1}\)

ĐK: x#0; x#-1

\(\frac{x^4}{1-x}\)+ x³ + x^{2 + 1}

^{= \(\frac{x^4}{1-x}\)}+ \(\frac{x^3\left(1-x\right)}{1-x}\)+ \(\frac{x^2\left(1-x\right)}{1-x}\)+ \(\frac{1-x}{1-x}\)

= \(\frac{x^4+x^3-x^4+x^2-x^3+1-x}{1-x}\)

= \(\frac{x+1}{1-x}\)

\(=\dfrac{2x-3y}{y\left(x-y\right)}+\dfrac{1}{x-y}=\dfrac{2x-3y+y}{y\left(x-y\right)}=\dfrac{2x-2y}{y\left(x-y\right)}=\dfrac{2}{y}\)

\(\dfrac{2x-3y}{xy-y^2}+\dfrac{1}{x-y}=\dfrac{2x-3y}{y\left(x-y\right)}+\dfrac{1}{x-y}=\dfrac{2x-3y+y}{y\left(x-y\right)}=\dfrac{2x-2y}{y\left(x-y\right)}=\dfrac{2}{y}.\)