Question

complete the factored form.

\\75x^2 + 20x - 7 = (5x - 1)( \quad )\\

\\75x^2 + 20x - 7 = (5x - 1)(\square)\\

Explanation:

⚡ Using what you learned: factoring trinomials

Step 1: Identify the missing terms

Let the missing binomial be \( (ax + b) \). We have:

$$ 75x^2 + 20x - 7 = (5x - 1)(ax + b) $$

Step 2: Determine the first term \( ax \)

Multiply the first terms of the binomials to get the leading term of the trinomial:

$$ 5x \cdot ax = 75x^2 $$

$$ a = \frac{75}{5} = 15 $$

So, the first term is \( 15x \).

Step 3: Determine the constant term \( b \)

Multiply the constant terms of the binomials to get the constant term of the trinomial:

$$ -1 \cdot b = -7 $$

$$ b = 7 $$

So, the constant term is \( 7 \).

Step 4: Verify the middle term

Multiply the binomials \( (5x - 1)(15x + 7) \) to check:

$$ (5x - 1)(15x + 7) = 75x^2 + 35x - 15x - 7 = 75x^2 + 20x - 7 $$

The terms match.

Answer:

\( 15x + 7 \)