4) \( 10 x^{2}-13 x+51 \) \( x^{4}-x^{3}-11 x^{2}-x-12 \) B) \( 2 x^{2}+3 \) \( x^{3}-2 x^{2}+x \) () \( \frac{x+4}{x^{3}-4 x^{2}+x+6} \)

Did you know that polynomial expressions can be classified based on their degrees? For instance, the first polynomial you provided, \( 10x^2 - 13x + 51 \), is a quadratic, which means its highest exponent is 2. Quadratics can show a range of behaviors, including having no real roots, one real root, or two real roots, depending on the discriminant \(b^2 - 4ac\). When simplifying rational functions, like \( \frac{x + 4}{x^3 - 4x^2 + x + 6} \), it's essential to factor the denominator when possible. Common mistakes include overlooking potential roots or failing to simplify further. Always remember to check for any common factors in the numerator and denominator to make your life easier, and ensure you're dealing with the simplest form of the expression!