Rewrite the expression as a product of four linear factors: (x^(2)+8x)^(2)+19(x^(2)+8x)+84 Answer:

Identify Expression: Let's first identify the expression we need to factor: The expression is (x^2 + 8x)^2 + 19(x^2 + 8x) + 84. We can notice that this is a quadratic in form, where the variable part (x^2 + 8x) is squared and then linearly added with a constant. Let's denote u = x^2 + 8x, so the expression becomes u^2 + 19u + 84.Factor Quadratic Expression: Now we need to factor the quadratic expression u^2 + 19u + 84. To do this, we look for two numbers that multiply to 84 and add up to 19. These numbers are 7 and 12. So we can write u^2 + 19u + 84 as (u + 7)(u + 12).Substitute Back and Simplify: Now we substitute back x^2 + 8x for u to get the factored form in terms of x: (u + 7)(u + 12) becomes (x^2 + 8x + 7)(x^2 + 8x + 12).Factor x^2 + 8x + 7: Next, we need to factor each of these quadratic expressions further. Starting with x^2 + 8x + 7, we look for two numbers that multiply to 7 and add up to 8. These numbers are 7 and 1. So we can write x^2 + 8x + 7 as (x + 7)(x + 1).Factor x^2 + 8x + 12: Now we factor x^2 + 8x + 12. We look for two numbers that multiply to 12 and add up to 8. These numbers are 6 and 2. So we can write x^2 + 8x + 12 as (x + 6)(x + 2).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 + 1)(x + 6)(x + 2).

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Rewrite the expression as a product of four linear factors:

(x^(2)+8x)^(2)+19(x^(2)+8x)+84
Answer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(x^{2}+8 x\right)^{2}+19\left(x^{2}+8 x\right)+84$ $\newline$ Answer:

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Evaluate definite integrals using the power rule

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

\newline

\left(x^{2}+8 x\right)^{2}+19\left(x^{2}+8 x\right)+84

\newline

Answer:

Identify Expression: Let's first identify the expression we need to factor: $\newline$ The expression is $(x^2 + 8x)^2 + 19(x^2 + 8x) + 84$ . $\newline$ We can notice that this is a quadratic in form, where the variable part $(x^2 + 8x)$ is squared and then linearly added with a constant. Let's denote $u = x^2 + 8x$ , so the expression becomes $u^2 + 19u + 84$ .
Factor Quadratic Expression: Now we need to factor the quadratic expression $u^2 + 19u + 84$ . To do this, we look for two numbers that multiply to $84$ and add up to $19$ . These numbers are $7$ and $12$ . So we can write $u^2 + 19u + 84$ as $(u + 7)(u + 12)$ .
Substitute Back and Simplify: Now we substitute back $x^2 + 8x$ for $u$ to get the factored form in terms of $x$ : $\newline$ $(u + 7)(u + 12)$ becomes $(x^2 + 8x + 7)(x^2 + 8x + 12)$ .
Factor $x^2 + 8x + 7$ : Next, we need to factor each of these quadratic expressions further. Starting with $x^2 + 8x + 7$ , we look for two numbers that multiply to $7$ and add up to $8$ . These numbers are $7$ and $1$ . So we can write $x^2 + 8x + 7$ as $(x + 7)(x + 1)$ .
Factor $x^2 + 8x + 12$ : Now we factor $x^2 + 8x + 12$ . We look for two numbers that multiply to $12$ and add up to $8$ . These numbers are $6$ and $2$ . So we can write $x^2 + 8x + 12$ as $(x + 6)(x + 2)$ .
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 + $1$ )(x + $6$ )(x + $2$ )\.