Factor the trinomial completely. 4q^2-11q-3 Select the correct choice below and, if necessary fill in the answer box within your choice. C A. 4q^2-11q-3=square B. The polynomial is prime

To factor the trinomial $4q^2 - 11q - 3$, we need to find two binomials that multiply to give this expression.

The general form for factoring a quadratic trinomial $ax^2 + bx + c$ is to find two numbers that multiply to $a \times c$ and add up to $b$.

1. Identify $a$, $b$, and $c$:
- $a = 4$
- $b = -11$
- $c = -3$

2. Calculate $a \times c = 4 \times (-3) = -12$.

3. Find two numbers that multiply to $-12$ and add up to $-11$. These numbers are $-12$ and $1$.

4. Rewrite the middle term $-11q$ using $-12$ and $1$:
$$4q^2 - 12q + q - 3$$

5. Factor by grouping:
- Group the terms: $(4q^2 - 12q) + (q - 3)$
- Factor out the greatest common factor from each group:
- From $4q^2 - 12q$, factor out $4q$: $4q(q - 3)$
- From $q - 3$, factor out $1$: $1(q - 3)$

6. Combine the factored groups:
$$4q(q - 3) + 1(q - 3) = (4q + 1)(q - 3)$$

Thus, the completely factored form of the trinomial is:

A. $4q^2 - 11q - 3 = (4q + 1)(q - 3)$