4. Calculate for which values of $x$ the value of the expression $2(x-1)^{2}+4$ is at least as much as the value of the expression $(x-1)(x+2)+x(x-3)$.

(6 points)
4. We set up the inequality $2(x-1)^{2}+4 \geq(x-1)(x+2)+x(x-3)$. Removing the parentheses, we get $2\left(x^{2}-2 x+1\right)+4 \geq x^{2}+x-2+x^{2}-3 x$ and rearrange the inequality $2 x^{2}-4 x+2+4 \geq 2 x^{2}-2 x-2$. Considering the rules for calculating with inequalities, we get $-2 x \geq-8$. Dividing by $-2$, we get $x \leq 4$.

Writing the inequality $2(x-1)^{2}+4 \geq(x-1)(x+2)+x(x-3) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-07.jpg?height=51&width=1636&top_left_y=1322&top_left_x=207)

Multiplication $(x-1)(x+2)=x^{2}+x-2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-07.jpg?height=57&width=1636&top_left_y=1422&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-07.jpg?height=48&width=1636&top_left_y=1475&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-07.jpg?height=49&width=1636&top_left_y=1529&top_left_x=210)