Rewrite the expression as a product of four linear factors: (3x^(2)+13 x)^(2)+18(3x^(2)+13 x)+56 Answer:

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Rewrite the expression as a product of four linear factors:  (3x^(2)+13 x)^(2)+18(3x^(2)+13 x)+56 Answer:

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Identify Given Expression: Identify the given expression and recognize that it resembles a quadratic in form, where the variable part is (3x^2 + 13x). The expression is: (3x^2 + 13x)^2 + 18(3x^2 + 13x) + 56Consider Quadratic Form: Notice that the expression can be considered as a quadratic equation in terms of (3x^2 + 13x). Let's denote u = 3x^2 + 13x. The expression becomes: u^2 + 18u + 56Factor Quadratic Expression: Factor the quadratic expression u^2 + 18u + 56. We are looking for two numbers that multiply to 56 and add up to 18. These numbers are 14 and 4. (u + 14)(u + 4) = u^2 + 18u + 56Substitute and Factor: Substitute back 3x^2 + 13x for u in the factored form. (3x^2 + 13x + 14)(3x^2 + 13x + 4)Factor First Quadratic: Now, we need to factor each quadratic separately. Starting with 3x^2 + 13x + 14, we look for two numbers that multiply to 3*14 = 42 and add up to 13. These numbers do not exist, so this quadratic does not factor over the integers. We must use the quadratic formula to find its roots.Apply Quadratic Formula: Apply the quadratic formula to 3x^2 + 13x + 14: x = [-13 ± sqrt(13^2 - 4*3*14)] / (2*3) x = [-13 ± sqrt(169 - 168)] / 6 x = [-13 ± sqrt(1)] / 6Solve for Roots: Solve for the two roots: x = (-13 + 1) / 6 and x = (-13 - 1) / 6 x = -12 / 6 and x = -14 / 6 x = -2 and x = -7/3First Linear Factors: Now we have two linear factors from the first quadratic: (x + 2) and (3x + 7)Factor Second Quadratic: Repeat the process for the second quadratic, 3x^2 + 13x + 4. We look for two numbers that multiply to 3*4 = 12 and add up to 13. These numbers do not exist, so this quadratic does not factor over the integers either. We must use the quadratic formula again.Apply Quadratic Formula: Apply the quadratic formula to 3x^2 + 13x + 4: x = [-13 ± sqrt(13^2 - 4*3*4)] / (2*3) x = [-13 ± sqrt(169 - 48)] / 6 x = [-13 ± sqrt(121)] / 6Solve for Roots: Solve for the two roots: x = (-13 + 11) / 6 and x = (-13 - 11) / 6 x = -2 / 6 and x = -24 / 6 x = -1/3 and x = -4Second Linear Factors: Now we have two more linear factors from the second quadratic: (x + 1/3) and (x + 4)Combine Linear Factors: Combine all four linear factors to express the original expression as a product of four linear factors: (x + 2)(3x + 7)(x + 1/3)(x + 4)

Rewrite the expression as a product of four linear factors: $\newline$ $\left(3 x^{2}+13 x\right)^{2}+18\left(3 x^{2}+13 x\right)+56$ $\newline$ Answer:

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Rewrite the expression as a product of four linear factors:\newline(3x2+13x)2+18(3x2+13x)+56 \left(3 x^{2}+13 x\right)^{2}+18\left(3 x^{2}+13 x\right)+56 (3x2+13x)2+18(3x2+13x)+56\newlineAnswer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(3 x^{2}+13 x\right)^{2}+18\left(3 x^{2}+13 x\right)+56$ $\newline$ Answer: