Can anyone explain this one ? , Thank you so much in advance ! <3?

For the equation 4.31x² + kx + 2.5 = 0 has no real roots,
Discriminant, Δ = k² - 4*4.31*2.5 < 0
k² - (√43.1)² < 0
(k + √43.1) (k - √43.1) < 0
-√43.1 < k < √43.1
Hence, min(k) ≈ -√43.1  and  max(k) ≈ √43.1

When k becomes k + n, the equation becomes 4.31x² + (k + n)x + 2.5 = 0
The new equation has two distinct real roots, and thus Δ > 0
(k + n)² - 4 * 4.31 * 2.5 > 0
(k + n)² - (√43.1)² > 0
[(k + n) + √43.1] [(k + n) - √43.1] > 0
k + n < -√43.1  or  k + n > √43.1
To ensure the above inequalities.
max(k) + n < -√43.1  or  min(k) + n > √43.1
√43.1 + n < -√43.1  or  -√43.1 + n > √43.1
n < -2√43.1  or  n > 2√43.1
n < -13.13  or  n > 13.13
Hence, Decrease k more than 13.13  or  Increase k more than 13.13

The answer: B) Decrease k by 15

The discriminant of the equation is
D=k^2-4(4.31)(2.5)=k^2-43.1
If no real root, D<0
=>
k^2<43.1
=>
|k|<6.565059025
If there are distinct real roots, then
|k|>6.565059025
If the original k=6.56059,say, then
6.56059-15=-8.43941
=>
|-8.43951|>6.565059025
Thus,
k should be decreased by 15. (B)

The discriminant of the equation 4.31x^2 +kx + 2.5 = 0 is k^2 -4*(4.31)(2.5) =
k^2 - 43.1. In order that equation has 2 distinct real roots we require that k^2 be
greater than 43.1, ie., require k less than -rt43.1 or k greater than rt43.1, ie., ;
require k less than -6.565059025 or greater than +6.565059905;
To guarantee equation has 2 distinct real roots (*) we have determined that the
absolute value of k must be greater than 6.565059905. Since the current value
of k was chosen to guarantee that the equation has no real roots, we know that
k lies within J = (-6.565059905,+6.565059905).;
Suppose k is in pos. half of J. Then only (B) guarantees (*).;
Suppose k is in neg. half of J. Then both (A) & (B) guarantee (*).;
No matter which half of J, k is in, (B) is the only move that will guarantee (*).

Bill had a good answer that covered the important facts about k and roots, but it didn't quite finish the problem correctly.

I think the question is clear. As Bill correctly points out, you have:
–6.565059 < k < 6.565059
since the equation has no real roots.

To make the equation have two distinct roots, you have to move k outside that range. Only B guarantees that. No matter what the initial value of k is, decreasing by 15 makes it < -6.565059. That's because even if k is the largest possible value, subtracting 15 makes it less than the smallest possible answer, and there will be two real roots.

For all the other answers, there is a value of k where the answer doesn't take it out of the range.  E.g., If k = -6.5, adding 13 get you to k = 6.5, and there are still no real roots.

y = 4.31x² + kx + 2.5

Discriminant
Δ = b²–4ac
Δ = 0, one real root –b/2a
Δ < 0, no real roots, two complex roots
Δ > 0, two real roots

Δ = b²–4ac = k²–4•4.31•2.5 = k²–43.1

for no real roots, Δ<0 or
k²–43.1 < 0
or
k² < 43.1
or
–6.565059 < k < 6.565059

to have 2 real roots Δ > 0
k²–43.1 > 0
–6.565059 > k > 6.565059

given the above:
the question is unclear. For example, k = 6.5 means no real roots, and k = 6.6 means two real roots, so increasing k by 0.1 makes the change.

but it depends on the initial value of k, which is unknown. It could be –6 which means an increase of 13 is needed. 

perhaps it means worse case?
then 13 is the answer, actually 13.13012

It will have two real roots if the quadratic discriminant (b^2 - 4ac) is positive.
It will have one real root if the quadratic discriminant is zero
it will have no real roots if the quadratic discriminant is negative.

The quadratic discriminant is k^2 - 4 (4.31)(2.5) = k^2 - 43.1

So for two real roots, k must be greater than 6.56