4. Find the equation of a line $L$ such that the graph of the function

$$
y=x^{4}+4 x^{3}-26 x^{2}
$$

lies on one side of this line, having two common points with it.
Answer: $y=60 x-225$.

Solution. Let $y=a x+b$ be the equation of the line $L$. We need to find such $a$ and $b$ that the equation

$$
x^{4}+4 x^{3}-26 x^{2}-(a x+b)=0
$$

has two roots $x_{1}$ and $x_{2}$ of even multiplicity. For this, the polynomial of the fourth degree must have the following factorization:

$$
x^{4}+4 x^{3}-26 x^{2}-a x-b=\left(x-x_{1}\right)^{2}\left(x-x_{2}\right)^{2} .
$$

Expanding the brackets on the right side and equating the coefficients of $x^{3}$ and $x^{2}$ on both sides of the identity, we get:

$$
-2\left(x_{1}+x_{2}\right)=4 ; \quad x_{2}^{2}+4 x_{1} x_{2}+x_{1}^{2}=-26 .
$$

From this,

$$
x_{1}+x_{2}=-2 ; \quad\left(x_{1}+x_{2}\right)^{2}+2 x_{1} x_{2}=-26 ; \quad x_{1} x_{2}=-15 .
$$

If $x_{1} Variant 1

## physics

