Polynomials and fractions

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. \[ \frac{mn} {m^{2} + 2mn + n^{2}} = \frac{*} {2m(m + n)^{3}} \]

Find the common denominator of the following three fractions. \[ \text{$ \frac{a} {a^{2}-ab}\ ,\qquad \frac{-b} {a^{2}-b^{2}} \ ,\qquad \frac{2b} {ab+b^{2}} $} \]

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