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Rewrite the expression as a product of four linear factors:

(x^(2)+5x)^(2)+10(x^(2)+5x)+24
Answer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(x^{2}+5 x\right)^{2}+10\left(x^{2}+5 x\right)+24$ $\newline$ Answer:

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Change of base formula

Full solution

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

\newline

\left(x^{2}+5 x\right)^{2}+10\left(x^{2}+5 x\right)+24

\newline

Answer:

Identify Trinomial: Identify the given expression as a perfect square trinomial. $\newline$ The expression $(x^2 + 5x)^2 + 10(x^2 + 5x) + 24$ is in the form of $(a + b)^2 = a^2 + 2ab + b^2$ , where $a = (x^2 + 5x)$ and $b$ is to be determined.
Determine Value of b: Determine the value of $b$ that makes the expression a perfect square trinomial. For the expression to be a perfect square trinomial, we need $b^2 = 24$ and $2ab = 10(x^2 + 5x)$ . Solving for $b$ , we get $b = \sqrt{24} = 2\sqrt{6}$ . Now we check if $2ab = 10(x^2 + 5x)$ holds true for $b = 2\sqrt{6}$ .
Verify $2ab$ Calculation: Verify that $2ab$ equals the middle term of the trinomial. $2ab = 2 \times (x^2 + 5x) \times 2\sqrt{6} = 4\sqrt{6}(x^2 + 5x)$ . We need this to equal $10(x^2 + 5x)$ . However, $4\sqrt{6}$ is not equal to $10$ , which means there is a mistake in the assumption that the expression is a perfect square trinomial.

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