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Rewrite the expression as a product of four linear factors:

(x^(2)+7x)^(2)+4(x^(2)+7x)-96
Answer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(x^{2}+7 x\right)^{2}+4\left(x^{2}+7 x\right)-96$ $\newline$ Answer:

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Identify factors

Full solution

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

\newline

\left(x^{2}+7 x\right)^{2}+4\left(x^{2}+7 x\right)-96

\newline

Answer:

Recognize Structure: Recognize the structure of the given expression The given expression resembles a quadratic in form, where the variable part $(x^2 + 7x)$ is squared and then linearly combined with a constant. This suggests that we might be able to factor it by treating $(x^2 + 7x)$ as a single variable.
Substitute Single Variable: Substitute a single variable for the repeated expression $\newline$ Let $y = x^2 + 7x$ . The expression becomes $y^2 + 4y - 96$ .
Factor Quadratic Expression: Factor the quadratic expression in terms of $y$ We need to find two numbers that multiply to $-96$ and add to $4$ . These numbers are $12$ and $-8$ . So, $y^2 + 4y - 96 = (y + 12)(y - 8)$ .
Substitute Back: Substitute back $x^2 + 7x$ for $y$ $\newline$ Replace $y$ with $x^2 + 7x$ in the factored form to get: $\newline$ $(x^2 + 7x + 12)(x^2 + 7x - 8)$ .
Factor Each Quadratic: Factor each quadratic separately $\newline$ We now have two quadratics to factor. We need to find two numbers that multiply to $12$ and add to $7$ for the first quadratic, and two numbers that multiply to $-8$ and add to $7$ for the second quadratic. $\newline$ For the first quadratic: The numbers are $3$ and $4$ . $\newline$ So, $x^2 + 7x + 12 = (x + 3)(x + 4)$ . $\newline$ For the second quadratic: The numbers are $1$ and $-8$ . $\newline$ So, $x^2 + 7x - 8 = (x + 8)(x - 1)$ .
Combine Factors: Combine the factors to express the original expression as a product of four linear factors $\newline$ The original expression is now factored as: $\newline$ $(x + 3)(x + 4)(x + 8)(x - 1)$ .

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