Question: Question 1 (5 points)\n\nWhat are the factors of the function:\n\n$x^2 + 8x + 7$\n\na $(x + 1)(x + 7)$\nb $(x + 5)(x + 1)$\nc $(x + 8)(x + 7)$\nd $(x + 2)(x + 3)$\n\nQuestion 2 (5 points)\n\nWhat are the solutions to the function:\n\n$x^2 + 8x + 7$\n\nx = __________\nx = __________\n\nBlank 1:\nBlank 2:
1. **State the problem:** We need to factor the quadratic function $$x^2 + 8x + 7$$ and find its solutions.
2. **Factoring the quadratic:** The general form is $$ax^2 + bx + c$$. Here, $$a=1$$, $$b=8$$, and $$c=7$$.
3. We look for two numbers that multiply to $$c=7$$ and add to $$b=8$$.
4. The pairs of factors of 7 are (1,7) and (-1,-7). Since $$1 + 7 = 8$$, the factors are $$1$$ and $$7$$.
5. Therefore, the factorization is:
$$x^2 + 8x + 7 = (x + 1)(x + 7)$$
6. **Answer to Question 1:** Option a: $(x + 1)(x + 7)$.
7. **Finding the solutions:** Set the factored form equal to zero:
$$ (x + 1)(x + 7) = 0 $$
8. By the zero product property, either:
$$ x + 1 = 0 \quad \Rightarrow \quad x = -1 $$
or
$$ x + 7 = 0 \quad \Rightarrow \quad x = -7 $$
9. **Answer to Question 2:**
$$ x = -1 $$
$$ x = -7 $$