Factor the following quadratic expressions, if possible. Some will not factor; if they don't factor, state that it does not factor. \( \begin{array}{ll}\text { 15. } x^{2}-5 x+6 & \text { 16. } x^{2}-7 x+6 \\ \text { 17. } x^{2}-5 x-36 & \text { 18. } m^{2}+16 m+63 \\ \text { 19. } s^{2}-3 s-1 & \text { 20. } x^{2}+7 x+2 \\ \text { 21. } x^{2}+14 x+49 & \text { 22. } x^{2}-9 \\ \text { 23. } c^{2}+11 c+3 & \\ \text { 24. Which quadratic expression above could } \\ \text { represent the area of a square? Explain. } \\ \text { 25. Would any of the expressions above NOT be the } \\ \text { side lengths for a rectangle? Explain. }\end{array} \)

The expression \(15. \, x^{2}-5 x+6\) does not factor nicely since its discriminant (\(b^2 - 4ac\)) is negative, leading to no real roots. For \(16. \, x^{2}-7 x+6\), this can be factored into \((x-6)(x-1)\). Next, \(17. \, x^{2}-5 x-36\) factors to \((x-9)(x+4)\), while \(18. \, m^{2}+16 m+63\) does not factor as there are no two numbers that multiply to 63 and add up to 16. The expression \(19. \, s^{2}-3 s-1\) does not factor over the reals due to its negative discriminant. As for \(20. \, x^{2}+7 x+2\), it does not factor nicely and needs the quadratic formula for roots. \(21. \, x^{2}+14 x+49\) factors perfectly as \((x+7)(x+7)\) or \((x+7)^2\), representing a perfect square. The expression \(22. \, x^{2}-9\) factors as \((x-3)(x+3)\). In \(23. \, c^{2}+11 c+3\), it does not factor cleanly due to a negative discriminant, indicating no real factored form. For \(24\), the quadratic expression that could represent the area of a square is \(21. \, x^{2}+14 x+49\), because it is a perfect square. The explanation lies in the fact that the area of a square is always represented by the square of a binomial. As for \(25\), any quadratic with a negative discriminant (like \(15\), \(18\), \(19\), \(20\), and \(23\)) would not represent the side lengths of a rectangle, since lengths must be non-negative in the real number system, and negative roots imply impossible dimensions.