Question

factor the trinomial completely.

\\20x^3 + 38x^2 + 12x\\

recall that the first step in factoring any polynomial is to look for a common factor.
notice that all three terms have a common factor of \\(2x\\). thus, factor it out.

\\20x^3 + 38x^2 + 12x = 2x(\quad)\\

Explanation:

⚡ Using what you learned: factoring trinomials · factoring monomials

Step 1: Factor out the greatest common factor

Divide each term of the polynomial by the greatest common factor, \(2x\):

$$ \frac{20x^3}{2x} = 10x^2 $$

$$ \frac{38x^2}{2x} = 19x $$

$$ \frac{12x}{2x} = 6 $$

Rewrite the expression:

$$ 20x^3 + 38x^2 + 12x = 2x(10x^2 + 19x + 6) $$

Step 2: Factor the remaining trinomial

To factor \(10x^2 + 19x + 6\), find two numbers that multiply to \(10 \times 6 = 60\) and add to \(19\). These numbers are \(15\) and \(4\).

Split the middle term:

$$ 10x^2 + 15x + 4x + 6 $$

Factor by grouping:

$$ 5x(2x + 3) + 2(2x + 3) = (5x + 2)(2x + 3) $$

Step 3: Write the completely factored expression

Combine the greatest common factor from Step 1 with the factored binomials from Step 2:

$$ 2x(5x + 2)(2x + 3) $$

Answer:

The missing expression in the box is:

$$ 10x^2 + 19x + 6 $$

The completely factored polynomial is:

$$ 2x(5x + 2)(2x + 3) $$