All pairs listed multiply to 48, hence they are valid pairs.

The other number is 72 divided by 8, which equals 9.

For the product to be a multiple of 15, at least one of the numbers must be a multiple of 5.

Using the fact that the product of the roots is c/a and the sum is -b/a, we can set a = 1, b = -1, c = -6.

By Vieta's formulas, the sum of the roots p + q = -b/a.

According to Vieta's formulas, the sum of the roots p and q is -b/a and the product is c/a.

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 => 4p^2 - 4p^2 + 16 >= 0, which is always true. Thus, p can be any real number.

Setting the discriminant to zero: (2p)^2 - 4(1)(p^2 - 4) = 0 leads to p = ±2.

The equation can be factored as (x + 1)(x + 1) = 0, giving the root -1 with multiplicity 2.

The discriminant is 2^2 - 4*1*1 = 0, indicating that the roots are real and equal.

The discriminant is 0, indicating that the roots are real and equal.

The equation can be factored as (x + 1)^2 = 0, giving the root x = -1.

For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0, leading to k = 1.

For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0, hence k > 1.

For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0 => 4 - 4k < 0 => k > 1.

Substituting x = -1 into the equation gives (-1)^2 + 2(-1) + k = 0, leading to 1 - 2 + k = 0, thus k = 1.

The product of the roots is 1 * (-3) = -3, hence k = -3.

For both roots to be negative, k must be greater than 0.

For both roots to be negative, k must be greater than 0.

For both roots to be positive, k must be less than 4 (from the condition of the sum and product of roots).

For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0 leads to k = -1.

The discriminant is 0, indicating one real root (a repeated root).

If one root is -2, then substituting x = -2 gives: (-2)^2 + 4(-2) + c = 0 => 4 - 8 + c = 0 => c = 4.

Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.

The product of the roots is given by c/a = 6/1 = 6.

Substituting x = 2 into the equation gives 2^2 + 5(2) + k = 0, leading to 4 + 10 + k = 0, thus k = -14.

The discriminant is 0, indicating one real root (a repeated root).

The discriminant is 0, indicating that the roots are real and equal.

Using Vieta's formulas, k = (-2)(-4) = 8.

For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.