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Rewrite the expression as a product of four linear factors:

(x^(2)+5x)^(2)-20(x^(2)+5x)+84
Answer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(x^{2}+5 x\right)^{2}-20\left(x^{2}+5 x\right)+84$ $\newline$ Answer:

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Complex conjugate theorem

Full solution

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

\newline

\left(x^{2}+5 x\right)^{2}-20\left(x^{2}+5 x\right)+84

\newline

Answer:

Recognize Quadratic Form: Let's first recognize that the given expression is a quadratic in form, where the variable is not just $x$ , but $(x^2 + 5x)$ . We can rewrite the expression as a quadratic equation: $\newline$ Let $u = x^2 + 5x$ . Then the expression becomes: $\newline$ $u^2 - 20u + 84$
Factor Quadratic Expression: Now, we need to factor the quadratic expression $u^2 - 20u + 84$ . To do this, we look for two numbers that multiply to $84$ and add up to $-20$ . These numbers are $-14$ and $-6$ . So we can write the factored form as: $(u - 14)(u - 6)$
Substitute Back and Simplify: Next, we substitute back $x^2 + 5x$ for $u$ in each factor to get the expression in terms of $x$ : $(x^2 + 5x - 14)(x^2 + 5x - 6)$
Factor First Quadratic Expression: Now, we need to factor each quadratic expression further. Starting with $x^2 + 5x - 14$ , we look for two numbers that multiply to $-14$ and add up to $5$ . These numbers are $7$ and $-2$ . So we can write the factored form as: $(x + 7)(x - 2)$
Factor Second Quadratic Expression: Next, we factor the second quadratic expression $x^2 + 5x - 6$ . We look for two numbers that multiply to $-6$ and add up to $5$ . These numbers are $6$ and $-1$ . So we can write the factored form as: $(x + 6)(x - 1)$
Combine Linear Factors: Finally, we combine all the linear factors to express the original expression as a product of four linear factors: x + \(7)(x - $2$ )(x + $6$ )(x - $1$ )\

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