Equation 1.


    I will derive this equation from the condition that the point (-2,79) lies on the graph, i.e. satisfies the equation

    a*(-2)^3 + b*(-2)^2  + c*(-2) + d = 79,   or

    -8a + 4b -2c + d = 79.          (1)



Equation 2.


    I will derive this equation from the condition that the point (1,-20) lies on the graph, i.e. satisfies the equation

    a*1^3 + b*1^2  + c*1 + d = -20,   or

    a + b + c + d = -20.            (2)



Equation 3.


    I will derive this equation from the condition that the slope at the point (-2,79) is -24.

    Since the first derivative is y'(x) = 3ax^2 + 2bx + c,  the equation for the slope is

    3a*(-2)^2 + 2b*(-2)  + c = -24,   or

    12a - 4b + c = -24.             (3)



Equation 4.


    I will derive this equation from the condition that the slope at the point (1,-20) is -24.

    Similarly to equation 3 case,  the equation for the slope is

    3a*1^2 + 2b*1  + c = -24,   or

    3a + 2b + c = -24.               (4)



So, your system of equations is  

     -8a + 4b -2c + d =  79         (1)
       a +  b + c + d = -20         (2)
     12a - 4b + c     = -24         (3)
      3a + 2b + c     = -24.        (4)


Next, use some technique or technology to solve it.

Probably, your hand calculator may help.


I use an online free of charge matrix equation solver

https://matrix.reshish.com/gauss-jordanElimination.php


I recommend you to get familiar with it.


To complete your assignment, input the augmented matrix into the solver and press the "Solve" button.