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Rewrite the expression as a product of four linear factors:

(x^(2)+x)^(2)-22(x^(2)+x)+40
Answer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(x^{2}+x\right)^{2}-22\left(x^{2}+x\right)+40$ $\newline$ Answer:

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Evaluate an exponential function

Full solution

Q. Rewrite the expression as a product of four linear factors:

\newline

\left(x^{2}+x\right)^{2}-22\left(x^{2}+x\right)+40

\newline

Answer:

Identify Expression: Let's first identify the expression we need to factor: $\newline$ The expression given is $(x^2 + x)^2 - 22(x^2 + x) + 40$ . $\newline$ Notice that this is a quadratic in form, where $(x^2 + x)$ is like a single variable. Let's substitute $u$ for $(x^2 + x)$ to make it easier to see the quadratic form. $\newline$ So, we have $u^2 - 22u + 40$ .
Substitute Variable: Now, we need to factor the quadratic expression $u^2 - 22u + 40$ . We are looking for two numbers that multiply to $40$ and add up to $-22$ . These numbers are $-20$ and $-2$ . So, we can write $u^2 - 22u + 40$ as $(u - 20)(u - 2)$ .
Factor Quadratic Expression: Next, we substitute back $(x^2 + x)$ for $u$ in the factored form to get $((x^2 + x) - 20)((x^2 + x) - 2)$ .
Substitute Back: Now, we need to factor each quadratic expression further. Starting with $(x^2 + x - 20)$ , we look for two numbers that multiply to $-20$ and add up to $1$ (the coefficient of $x$ ). These numbers are $5$ and $-4$ . So, we can write $x^2 + x - 20$ as $(x + 5)(x - 4)$ .
Factor Further: Similarly, we factor $x^2 + x - 2$ by finding two numbers that multiply to $-2$ and add up to $1$ . These numbers are $2$ and $-1$ . So, we can write $x^2 + x - 2$ as $(x + 2)(x - 1)$ .
Combine Linear Factors: Finally, we combine all the linear factors to express the original expression as a product of four linear factors: x + \(5)(x - $4$ )(x + $2$ )(x - $1$ )\.

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