Question

factor the trinomial by grouping. 8x² + 14x + 3
a. find two numbers whose product is 8·3 = 24 and whose sum is 14.
b. write 14x using the factors from part (a).
c. factor by grouping.
a. the two numbers with a product of 24 and a sum of 14 are 2,12. (use a comma to separate answers as needed.)
b. 8x² + 14x + 3 = 8x²+ + 3

Explanation:

Step1: Recall the two - numbers result

We found the two numbers 2 and 12 in part (a). We rewrite 14x as the sum of terms using these two numbers. Since 14x=2x + 12x, we have:
$8x^{2}+14x + 3=8x^{2}+2x+12x + 3$

Step2: Factor by grouping

Group the first two terms and the last two terms:
$(8x^{2}+2x)+(12x + 3)$
Factor out the greatest - common factor from each group. From $8x^{2}+2x$, the GCF is 2x, so $8x^{2}+2x=2x(4x + 1)$. From $12x + 3$, the GCF is 3, so $12x + 3=3(4x + 1)$.
Then $(8x^{2}+2x)+(12x + 3)=2x(4x + 1)+3(4x + 1)=(4x + 1)(2x+3)$

Answer:

$(4x + 1)(2x + 3)$