Rewrite the expression as a product of four linear factors: (x^(2)+6x)^(2)-11(x^(2)+6x)-80 Answer:

Simplify Expression: Let's first simplify the given expression: (x^2 + 6x)^2 - 11(x^2 + 6x) - 80 We can let y = x^2 + 6x to make the expression look like a quadratic in terms of y: y^2 - 11y - 80Substitute and Factor: Now we factor the quadratic expression: y^2 - 11y - 80 = (y - 16)(y + 5)Factor Quadratic Expression: Next, we substitute back x^2 + 6x for y: (x^2 + 6x - 16)(x^2 + 6x + 5)Factor First Quadratic: We now factor each quadratic expression to find the linear factors. Starting with x^2 + 6x - 16, we look for two numbers that multiply to -16 and add up to 6. These numbers are 8 and -2. x^2 + 6x - 16 = (x + 8)(x - 2)Factor Second Quadratic: Next, we factor x^2 + 6x + 5, looking for two numbers that multiply to 5 and add up to 6. These numbers are 5 and 1. x^2 + 6x + 5 = (x + 5)(x + 1)Write as Product: Finally, we write the expression as a product of four linear factors: (x + 8)(x - 2)(x + 5)(x + 1)

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

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\left(x^{2}+6 x\right)^{2}-11\left(x^{2}+6 x\right)-80

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Answer:

Simplify Expression: Let's first simplify the given expression: $\newline$ $(x^2 + 6x)^2 - 11(x^2 + 6x) - 80$ $\newline$ We can let $y = x^2 + 6x$ to make the expression look like a quadratic in terms of $y$ : $\newline$ $y^2 - 11y - 80$
Substitute and Factor: Now we factor the quadratic expression: $\newline$ $y^2 - 11y - 80 = (y - 16)(y + 5)$
Factor Quadratic Expression: Next, we substitute back $x^2 + 6x$ for $y$ : $(x^2 + 6x - 16)(x^2 + 6x + 5)$
Factor First Quadratic: We now factor each quadratic expression to find the linear factors. Starting with $x^2 + 6x - 16$ , we look for two numbers that multiply to $-16$ and add up to $6$ . These numbers are $8$ and $-2$ . $\newline$ $x^2 + 6x - 16 = (x + 8)(x - 2)$
Factor Second Quadratic: Next, we factor $x^2 + 6x + 5$ , looking for two numbers that multiply to $5$ and add up to $6$ . These numbers are $5$ and $1$ . $\newline$ $x^2 + 6x + 5 = (x + 5)(x + 1)$
Write as Product: Finally, we write the expression as a product of four linear factors: x + \(8)(x - $2$ )(x + $5$ )(x + $1$ )\