Rewrite the expression as a product of four linear factors: (4x^(2)+15 x)^(2)+20(4x^(2)+15 x)+99 Answer:

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Rewrite the expression as a product of four linear factors:  (4x^(2)+15 x)^(2)+20(4x^(2)+15 x)+99 Answer:

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Recognize perfect square trinomial: Recognize the given expression as a perfect square trinomial. The expression is in the form of (A^2 + 2AB + B^2), where A = (4x^2 + 15x) and B is a number we need to find such that B^2 = 99 and 2AB = 20(4x^2 + 15x).Find value of B: Find the value of B. We know that B^2 = 99, so B is the square root of 99. Since 99 is not a perfect square, we can simplify it to B = √(9*11) = 3√11.Verify 2AB matches: Verify that 2AB matches the middle term of the original expression. 2AB = 2 * (4x^2 + 15x) * 3√11 = 20(4x^2 + 15x) after simplifying. This matches the middle term of the original expression, confirming that B = 3√11 is correct.Write as perfect square: Write the expression as a perfect square. The original expression can now be written as a perfect square: [(4x^2 + 15x) + 3√11]^2.Factor binomial squared: Factor the perfect square as a binomial squared. The expression [(4x^2 + 15x) + 3√11]^2 is the square of the binomial (4x^2 + 15x + 3√11).Factor quadratic term: Factor the quadratic term within the binomial. To factor 4x^2 + 15x + 3√11, we look for two numbers that multiply to 4*3√11 and add to 15. However, this is not possible with integers, so we need to use the quadratic formula to find the roots of 4x^2 + 15x + 3√11 = 0.Apply quadratic formula: Apply the quadratic formula to find the roots. The quadratic formula is x = [-b ± √(b^2 - 4ac)] / (2a), where a = 4, b = 15, and c = 3√11.Calculate discriminant: Calculate the discriminant (b^2 - 4ac). The discriminant is 15^2 - 4*4*3√11 = 225 - 48√11.Calculate roots: Calculate the roots using the quadratic formula. x = [-15 ± √(225 - 48√11)] / 8.Simplify roots: Simplify the roots. The roots cannot be simplified further without a calculator, but they represent the two linear factors of the quadratic term.Write final expression: Write the final expression as a product of linear factors. The final expression is the product of the binomial squared and its linear factors: (x - root1)(x - root2)(4x^2 + 15x + 3√11).Realize factorization mistake: Realize a mistake has been made in the factorization process. The original approach to factor the quadratic term within the binomial was incorrect because we cannot factor 4x^2 + 15x + 3√11 into linear factors with real coefficients. We need to go back and find the correct linear factors.

Rewrite the expression as a product of four linear factors: $\newline$ $\left(4 x^{2}+15 x\right)^{2}+20\left(4 x^{2}+15 x\right)+99$ $\newline$ Answer:

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Rewrite the expression as a product of four linear factors:\newline(4x2+15x)2+20(4x2+15x)+99 \left(4 x^{2}+15 x\right)^{2}+20\left(4 x^{2}+15 x\right)+99 (4x2+15x)2+20(4x2+15x)+99\newlineAnswer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(4 x^{2}+15 x\right)^{2}+20\left(4 x^{2}+15 x\right)+99$ $\newline$ Answer: