Question

1. factor: $3x^{2}-x - 14$

Explanation:

Step1: Multiply coefficients

For the quadratic expression $3x^{2}-x - 14$, multiply the coefficient of $x^{2}$ (which is 3) and the constant term (- 14). So, $3\times(-14)=-42$.

Step2: Find two - numbers

We need to find two numbers that multiply to - 42 and add up to the coefficient of $x$ (which is - 1). The numbers are 6 and - 7 since $6\times(-7)=-42$ and $6+( - 7)=-1$.

Step3: Rewrite the middle term

Rewrite the middle - term $-x$ as $6x-7x$. So, $3x^{2}-x - 14=3x^{2}+6x-7x - 14$.

Step4: Group the terms

Group the terms: $(3x^{2}+6x)-(7x + 14)$.

Step5: Factor out the greatest common factor from each group

From the first group $3x^{2}+6x$, the GCF is $3x$, so $3x^{2}+6x = 3x(x + 2)$. From the second group $7x + 14$, the GCF is 7, so $7x + 14=7(x + 2)$. Then we have $3x(x + 2)-7(x + 2)$.

Step6: Factor out the common binomial factor

Factor out the common binomial factor $(x + 2)$: $(3x-7)(x + 2)$.

Answer:

$(3x - 7)(x + 2)$