Question

factor the trinomial completely.

\\5m^2 + 14m + 15\\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. \\(5m^2 + 14m + 15 = \\) (factor completely.)
b. the polynomial is prime.

Explanation:

⚡ Using what you learned: factoring trinomials

Step 1: Identify the coefficients

For the trinomial \( 5m^2 + 14m + 15 \):

$$ a = 5, \quad b = 14, \quad c = 15 $$

Step 2: Find the product \( ac \) and list factors

$$ ac = 5 \times 15 = 75 $$

We need two integers that multiply to \( 75 \) and add up to \( 14 \).

Pairs of factors of \( 75 \):

• \( 1 \times 75 \implies 1 + 75 = 76 \)
• \( 3 \times 25 \implies 3 + 25 = 28 \)
• \( 5 \times 15 \implies 5 + 15 = 20 \)

Step 3: Determine if the polynomial is factorable

None of the factor pairs of \( 75 \) add up to \( 14 \). Since there are no integers that satisfy both conditions, the trinomial cannot be factored using integer coefficients.

Answer:

B. The polynomial is prime.