A family of cubic functions have zeros of -2 and 4±7. 4) Write the equation...

Step 1
Family of cubic functions that have the roots a,b and c is ${\displaystyle y=D(x-a)(x-b)(x-c)}$, here D is a constant.
Given roots are: ${\displaystyle a=-2,b=4+\sqrt{7}\text{and}c=4-\sqrt{7}}$.
(a) ${\displaystyle y=D(x-(-2))(x-(4+\sqrt{7}))(x-(4-\sqrt{7}))}$
${\displaystyle =D(x+2)((x-4)-\sqrt{7})((x-4)+\sqrt{7})}$
${\displaystyle =D(x+2)({(x-4)}^2-7)(A-B)(A+B)}$
${\displaystyle =A^2-B^2}$
${\displaystyle =D(x+2)(x^2+16-8x-7){(A-B)}^2=A^2+B^2-2AB}$
${\displaystyle =D(x+2)(x^2-8x+9)}$
Step 2
(b) Substitute ${\displaystyle x=2\text{and}y=-72\int oy=D(x+2)(x^2-8x+9)}$ and solve for D.
${\displaystyle -712=D(2+2)(2^2-8\left(2\right)+9)}$
${\displaystyle 6=D}$
For cubic equation which goes through point (2, -72) substitute ${\displaystyle D=6\oint y=D(x+2)(x^2-8+9)}$
we get, ${\displaystyle y=6(x+2)(x^2-8x+9)}$
The graph ${\displaystyle y=6(x+2)(x^2-8x+9)}$ shows the cubic equation passing through the point (2, -72) which have
the three roots ${\displaystyle -2,4+\sqrt{7}\approx 6.646\text{and}4-\sqrt{7}\approx 1.354}$.