Question

1. factor: 3x² - 5x - 12

Explanation:

Step1: Multiply coefficients

Multiply the coefficient of $x^{2}$ (which is 3) and the constant term (- 12). So, $3\times(-12)=-36$.

Step2: Find two - numbers

Find two numbers that multiply to - 36 and add up to the coefficient of $x$ (-5). The numbers are 4 and - 9 since $4\times(-9)=-36$ and $4+( - 9)=-5$.

Step3: Rewrite the middle term

Rewrite the quadratic expression $3x^{2}-5x - 12$ as $3x^{2}+4x-9x - 12$.

Step4: Group the terms

Group the terms: $(3x^{2}+4x)-(9x + 12)$.

Step5: Factor out the GCF from each group

From the first group $3x^{2}+4x$, the GCF is $x$, so we get $x(3x + 4)$. From the second group $9x + 12$, the GCF is 3, so we get $3(3x + 4)$.

Step6: Factor out the common binomial factor

Factor out the common binomial factor $(3x + 4)$: $(3x + 4)(x-3)$.

Answer:

$(3x + 4)(x - 3)$