Determine 2 values of k so that trinomials can be factored over integers 36m^2 + 8m + k and 18x^2 - 42y + k.

To determine two values of k so that the trinomials can be factored over integers, we need to find the factors of the leading coefficient (the coefficient of the squared term) multiplied by the constant term.

For the first trinomial, 36m^2 + 8m + k, the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of k (the constant term) can be positive or negative factors of k, so we include both positive and negative versions of these factors.

For the second trinomial, 18x^2 - 42y + k, the factors of 18 are: 1, 2, 3, 6, 9, and 18.
Similar to the first trinomial, we include both positive and negative factors of k.

Given these factors, we need to find two values for k where the factors allow the trinomials to be factored over integers.

Let's go through the options:

For the first trinomial:
If we consider the factors of 36m^2, we can try k = 1.
Factors of 1: ±1
If we substitute k = 1, the trinomial becomes: 36m^2 + 8m + 1
However, this trinomial cannot be factored over integers.

If we consider the factors of 18x^2, we can try k = 4.
Factors of 4: ±1, ±2, ±4
If we substitute k = 4, the trinomial becomes: 18x^2 - 42y + 4
By regrouping, this trinomial can be factored as: 2(9x^2 - 21y) + 4
Common factor: 2
Factoring the parenthetical expression: 2(3x - 7y)(3x - 7y)
So, for the first trinomial, k = 4 allows it to be factored over integers.

For the second trinomial:
If we consider the factors of 36m^2, we can also try k = 1.
Factors of 1: ±1
If we substitute k = 1, the trinomial becomes: 18x^2 - 42y + 1
However, this trinomial cannot be factored over integers.

If we consider the factors of 18x^2, we can try k = 9.
Factors of 9: ±1, ±3, ±9
If we substitute k = 9, the trinomial becomes: 18x^2 - 42y + 9
By regrouping, this trinomial can be factored as: 3(6x^2 - 14y) + 9
Common factor: 3
Factoring the parenthetical expression: 3(2x - 7y)(3x - 7y)
So, for the second trinomial, k = 9 allows it to be factored over integers.

In summary, the values of k that allow the trinomials to be factored over integers are k = 4 for 36m^2 + 8m + k and k = 9 for 18x^2 - 42y + k.