Answer:
See attachments.
Step-by-step explanation:
Question 1
To factor the quadratic expression x² + 8x + 15, we first need to identify the pairs of factors of the product of the coefficient of x² and the constant.
In this case, the coefficient of x² is one and the constant is 15, so their product is 1 × 15 = 15. Therefore, we need to list pairs of factors of 15, which are:
Next, we need to identify the pair of factors that has a sum equal to the coefficient of the x-term in the quadratic expression. In this case, the coefficient of the x-term is 8, so we need to identify the pair that has a sum of 8. The pair of factors that has a sum of 8 is:
To write the quadratic expression as the product of two binomials, rewrite the middle term as the sum of 3 and 5:
\(x^2+3x+5x+15\)
Now, factor the first two terms and the last two terms separately:
\(x(x+3)+5(x+3)\)
Finally, factor out the common term (x + 3):
\((x+3)(x+5)\)
Therefore, the given quadratic expression written as a product of two binomials is:
\(\Large\boxed{\boxed{(x+3)(x+5)}}\)
\(\dotfill\)
Question 2
To factor the quadratic expression x² + 2x - 15, we first need to identify the pairs of factors of the product of the coefficient of x² and the constant.
In this case, the coefficient of x² is one and the constant is -15, so their product is 1 × 15 = -15. Therefore, we need to list pairs of factors of -15, which are:
- -1 and 15
- 1 and -15
- -3 and 5
- 3 and -5
Next, we need to identify the pair of factors that has a sum equal to the coefficient of the x-term in the quadratic expression. In this case, the coefficient of the x-term is 2, so we need to identify the pair that has a sum of 2. The pair of factors that has a sum of 2 is:
To write the quadratic expression as the product of two binomials, rewrite the middle term as the sum of -3 and 5:
\(x^2-3x+5x-15\)
Now, factor the first two terms and the last two terms separately:
\(x(x-3)+5(x-3)\)
Finally, factor out the common term (x - 3):
\((x-3)(x+5)\)
Therefore, the given quadratic expression written as a product of two binomials is:
\(\Large\boxed{\boxed{(x-3)(x+5)}}\)
