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1

Solved example of Square of a trinomial

$x\left(\left(1+x\right)\left(x-1\right)+8\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
2

Solve the product $\left(1+x\right)\left(x-1\right)$

$x\left(x^2-11^2+8\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
3

Calculate the power

$x\left(x^2-1\cdot 1+8\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
4

Apply the formula: $-x$, where $x=1$

$x\left(x^2-1+8\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
5

Subtract the values $8$ and $-1$

$x\left(x^2+7\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
6

Solve the product $x\left(x+2\right)$

$x\cdot x^2+7x+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x^2+2x-\left(x-3\right)^2$
7

When multiplying exponents with same base you can add the exponents

$x^{3}+7x+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x^2+2x-\left(x-3\right)^2$
8

Grouping terms

$x^{3}+7x+\left(-x^2-x-1\right)^2-x\left(1+x\right)^3-x^2-2x+1\left(x-3\right)^2=0$
9

Any expression multiplied by $1$ is equal to itself

$x^{3}+7x+\left(-x^2-x-1\right)^2-x\left(1+x\right)^3-x^2-2x+\left(x-3\right)^2=0$
10

Expand $\left(x-3\right)^2$

$x^{3}+7x+\left(-x^2-x-1\right)^2-x\left(1+x\right)^3-x^2-2x+x^2-6x+9=0$
11

Moving the term $9$ to the other side of the equation with opposite sign

$x^{3}+7x+\left(-x^2-x-1\right)^2-x\left(1+x\right)^3-x^2-2x+x^2-6x=-9$
12

Expand the trinomial using the formula $\left(a+b+c\right)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$

$x^{3}+7x+x^{4}+\left(-x\right)^2+1+2x^2x+2x^2+2x-x\left(1+x\right)^3-x^2-2x+x^2-6x=-9$
13

When multiplying exponents with same base you can add the exponents

$x^{3}+7x+x^{4}+\left(-x\right)^2+1+2x^{3}+2x^2+2x-x\left(1+x\right)^3-x^2-2x+x^2-6x=-9$
14

The power of a product is equal to the product of it's factors raised to the same power

$x^{3}+7x+x^{4}+x^2+1+2x^{3}+2x^2+2x-x\left(1+x\right)^3-x^2-2x+x^2-6x=-9$
15

Adding $2x^{3}$ and $x^{3}$

$7x+x^{4}+x^2+1+2x^2+2x-x\left(1+x\right)^3-x^2-2x+x^2-6x+3x^{3}=-9$
16

Adding $x^2$ and $x^2$

$7x+x^{4}+1+2x^2+2x-x\left(1+x\right)^3-x^2-2x-6x+3x^{3}+2x^2=-9$
17

Adding $2x^2$ and $-1x^2$

$x^{4}+1-x\left(1+x\right)^3-2x-6x+3x^{3}+2x^2+9x+x^2=-9$
18

Moving the term $1$ to the other side of the equation with opposite sign

$x^{4}-x\left(1+x\right)^3-2x-6x+3x^{3}+2x^2+9x+x^2=-10$
19

Apply the rule of the cube of a binomial

$x^{4}-x\left(1+x\right)\left(1+2x+x^2\right)-2x-6x+3x^{3}+2x^2+9x+x^2=-10$
20

Adding $2x^2$ and $x^2$

$x^{4}-x\left(1+x\right)\left(1+2x+x^2\right)-2x-6x+3x^{3}+9x+3x^2=-10$
21

Solve the product $x\left(2x+x^2\right)$

$x^{4}+\left(-1-x\right)\left(x+2x^2+x^{3}\right)-2x-6x+3x^{3}+9x+3x^2=-10$
22

Adding $-6x$ and $9x$

$x^{4}+\left(-1-x\right)\left(x+2x^2+x^{3}\right)-2x+3x^{3}+3x^2+3x=-10$
23

We can factor the polynomial $\left(x+2x^2+x^{3}\right)$ using synthetic division (Ruffini's rule). We found that $-1$ is a root of the polynomial

$-1+2{\left(-1\right)}^2+{\left(-1\right)}^{3}=0$
24

Let's divide the polynomial by $x+1$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $-1$. Add the result to the second coefficient and then multiply this by $-1$ and so on

$\left|\begin{array}{c}1 & 2 & 1 & 0 \\ & -1 & -1 & 0 \\ 1 & 1 & 0 & 0\end{array}\right|-1$
25

In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $x+1$

$x^{4}+\left(-1-x\right)\left(x+x^{2}\right)\left(x+1\right)-2x+3x^{3}+3x^2+3x=-10$
26

Multiplying polynomials $x$ and $-x^{2}x+-x^{2}$

$x^{4}-x^2-x-x\cdot x^2-x^2-x^{3}-x^{2}-x^{4}-x^{3}-2x+3x^{3}+3x^2+3x=-10$
27

When multiplying exponents with same base you can add the exponents

$x^{4}-x^2-x-x^{3}-x^2-x^{3}-x^{2}-x^{4}-x^{3}-2x+3x^{3}+3x^2+3x=-10$
28

Subtracting $x^{4}$ and $x^{4}$

$-x^2-x-x^{3}-x^2-x^{3}-x^{2}-x^{3}-2x+3x^{3}+3x^2+3x+0=-10$
29

Adding $-x^2$ and $-x^2$

$-x-x^{3}-x^{3}-x^{2}-x^{3}-2x+3x^{3}+3x^2+3x+0-2x^2=-10$
30

Adding $-2x^{3}$ and $2x^{3}$

$0+0x+0x^2+0x^{3}=-10$
31

Any expression multiplied by $0$ is equal to $0$

$0+0+0+0=-10$
32

Add the values $0$ and $0$

$0=-10$
33

$0$ not equal to $-10$

$false$

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