The roots of the equation 3x^2 + 5x + 1 =0 are a and b while the roots of the equation hx^2 - 4x + k =0 are a+3 and b+3. Find values of h&k?

a and b are the roots of 3x² + 5x + 1 = 0
Sum of roots: a + b = -5/3 ...... [1]
Product of roots: ab = 1/3 ...... [2]

(a + 3) and (b + 3) are the roots of hx² - 4x + k = 0

Sum of roots :
(a + 3) + (b + 3) = 4/h
4/h = (a + b) + 6 ...... [3]

Substitute [1] into [3] :
4/h = (-5/3) + 6
4/h = 13/3
13h = 12
h = 12/13

Product of roots :
(a + 3)(b + 3) = k/h
ab + 3a + 3b + 9 = k/h
ab + 3(a + b) + 9 = k/h ...... [4]

Substitute [1], [2] and h = 12/13 into [4] :
(1/3) + 3(-5/3) + 9 = k/(12/13)
13/3 = (13/12)k
(1/4)k = 1
k = 4

Answers : h = 12/13, k = 4

Dividing each quadratic by its leading coefficient, we have

x² + (5/3)x + 1/3
= (x - a)(x - b)
= x² + -(a+b)x + ab

and

x² + -(4/h)x + k/h
= (x - (a+3))(x - (b+3))
= x² + -(a+b+6)x + (ab+3(a+b)+9).

Matching coefficients in each, we get

a+b = -5/3
ab = 1/3

and

a+b+6 = 4/h
ab+3(a+b)+9 = k/h

respectively. We can use the first pair of equations to replace reach occurrence of a+b and ab in the second pair:

-5/3 + 6 = 4/h
1/3 + 3(-5/3) + 9 = k/h.

Solving the first of these for h yields

h = 12/13.

Plugging this in for h in the second and solving for k yields

k = 4.