step 1
we have the function
[tex]y=(x-2)(x^2+3x)[/tex]
Applying the distributive property
[tex]\begin{gathered} y=x^3+3x^2-2x^2-6x \\ y=x^3+x^2-6x \end{gathered}[/tex]
Find out the first derivative of the given function y
[tex]y^{\prime}=3x^2+2x-6[/tex]
Remember that the derivative of the function is the same that the slope
so
[tex]m=3x^2+2x-6[/tex]
Evaluate at the point (1,-4)
[tex]\begin{gathered} m=3(1)^2+2(1)-6 \\ m=3+2-6 \\ m=-1 \end{gathered}[/tex]
step 2
Find out the equation of the line in slope-intercept form
y=mx+b
we have
m=-1
point (1,-4)
substitute and solve for b
-4=-1(1)+b
b=-4+1
b=-3
therefore
The equation of the line is
y=-x-3
Verify with a graphing tool
