Can Someone Please Explain This Tangent/normal Answer

but with b, the answer says:

x^3+2x^2+2x-2=x-2
x^3+2x^2+x=0
x(x^2+2x+1)=0
x(x+1)^2=0
REPEATED ROOT THEREFORE TANGENT
point of contact where x=-1
THEREFORE (-1,-3)

Notice that x(x+1)^2=x(x+1)(x+1)=0. So, the only solution is for x=-1, i.e. a repeated root. So x=-1 is plugged into the prior equation to give y=-3.

the repeated root of the equation above is x = -1.
when a curve touches a line, say curve y=f(x) and line y=ax + b
you will have a repeated root for the equation f(x) = ax+b

repeated root means it only touches the line once. a quadratic with two distinct roots would cross twice because of its shape...

(you might have to bear with me in this explanation and use your imagination lol)

is that because the line y=ax+b is acting like the x axis does with a quadratic equation. When you have repeated roots in a quadratic, the x axis acts as a tangent.

Therefore when you say that f(x)=ax+b, you are simply putting the two together, to work this out, but instead of the x-axis as the tangent, you will be using the y=ax+b?

yeh ok ok!! lol, i meant a quadratic that touches it once! lol. Then would my reasoning be correct?