Factor completely: $\newline$ $(x-9)(3 x+5)+(4 x+9)(3 x+5)^{2}$ $\newline$ Answer:

Full solution

Q. Factor completely:

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(x-9)(3 x+5)+(4 x+9)(3 x+5)^{2}

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Answer:

Identify common factors: Identify common factors in both terms of the expression. $\newline$ The expression is $(x-9)(3x+5)+(4x+9)(3x+5)^{2}$ . We can see that $(3x+5)$ is a common factor.
Factor out common factor: Factor out the common factor $(3x+5)$ . We can write the expression as $(3x+5)[(x-9) + (4x+9)(3x+5)]$ .
Distribute inside the brackets: Distribute the $(3x+5)$ inside the brackets. $\newline$ Now we need to multiply $(4x+9)$ by $(3x+5)$ and add $(x-9)$ to it. This gives us $(3x+5)[(x-9) + (12x^2 + 27x + 45)]$ .
Combine like terms: Combine like terms inside the brackets. $\newline$ We combine $(x-9)$ with $(12x^2 + 27x + 45)$ to get $(3x+5)(12x^2 + 28x + 36)$ .
Check for further factoring: Check for any further factoring possibilities. $\newline$ The quadratic expression $12x^2 + 28x + 36$ can be factored further. We look for two numbers that multiply to $12\times36$ and add to $28$ . These numbers are $12$ and $24$ .
Factor quadratic expression: Factor the quadratic expression. $\newline$ We can write $12x^2 + 28x + 36$ as $(4x+6)(3x+6)$ . So the expression becomes $(3x+5)(4x+6)(3x+6)$ .
Check for common factors: Check for any common factors or simplifications. $\newline$ There are no common factors between $(3x+5)$ , $(4x+6)$ , and $(3x+6)$ , and no further simplifications can be made.