Question

the following factorization is incorrect. find the mistake, and correct the right-hand side.
$14x^2 + 12x + 8 = 2(7x + 6)$
$14x^2 + 12x + 8 = \square$
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\\(\boldsymbol{}\\) question 6 \\(\boldsymbol{}\\) 0/1 pt \\(\boldsymbol{}\\) 3 \\(\boldsymbol{}\\) 99 \\(\boldsymbol{}\\) details
factor the following trinomial completely. look first for the greatest common factor.
$6x^3 - x^2 - 35x$
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Explanation:

Problem 1: Fix incorrect factorization

Step1: Identify missing term in factoring

The original right-hand side omitted the $x^2$ term and the constant term after factoring out GCF 2.
Factor out GCF 2 from each term:
$14x^2 + 12x + 8 = 2(7x^2) + 2(6x) + 2(4)$

Step2: Combine factored terms

Group the factored terms inside parentheses:
$2(7x^2 + 6x + 4)$

Problem 2: Factor trinomial completely

Step1: Factor out GCF $x$

The GCF of $6x^3$, $-x^2$, $-35x$ is $x$:
$6x^3 - x^2 - 35x = x(6x^2 - x - 35)$

Step2: Factor quadratic trinomial

Factor $6x^2 - x - 35$ by finding two numbers that multiply to $6\times(-35)=-210$ and add to $-1$ (these numbers are $14$ and $-15$):
$6x^2 +14x -15x -35 = 2x(3x+7) -5(3x+7) = (2x-5)(3x+7)$

Step3: Combine all factors

Multiply the GCF with the factored quadratic:
$x(2x-5)(3x+7)$

Answer:

1. $2(7x^2 + 6x + 4)$
2. $x(2x-5)(3x+7)$