Polynomials and Rational Expressions

Assuming \(x\in \mathbb{R}\setminus \left \{0, 1, 3\right \}\), find the quotient of the polynomials. \[ (-5x^{4} + 4x^{2} + 3x - 4) : (x^{3} - 4x^{2} + 3x) \]

Identify a correct expression which should be placed to the starred position in the following expression to obtain a valid statement. \[ \frac{mn} {m^{2} + 2mn + n^{2}} = \frac{*} {2m(m + n)^{3}} \]

Assuming \(x\neq 0\), \(x\neq \pm 1\), \(y\neq 0\), simplify the expression. \[ \left [\left ( \frac{x} {x + 1}\right )^{2} : \left (\frac{x - 1} {y} \right )^{2}\right ] : \frac{2xy} {x^{2} - 1} \]

Assuming \(x\neq \pm y\) and \(y\neq 2x\), simplify the expression: \[ \left ( \frac{2x} {x + y} + \frac{y} {x - y} - \frac{y^{2}} {x^{2} - y^{2}}\right ) : \left ( \frac{1} {x + y} + \frac{x} {x^{2} - y^{2}}\right ) \]