The problem in the image is asking about a quadratic function $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. The graph passes through the points $(7, 0)$ and $(-3, 0)$. We are tasked with finding the value of $a + b$, given that $a$ is an integer greater than 1.

Steps:

1. Roots of the Quadratic: Since $(7, 0)$ and $(-3, 0)$ are points on the graph, they are the roots of the quadratic equation. Therefore, the quadratic can be written in factored form: $f(x) = a(x - 7)(x + 3)$ Expanding this expression gives: $f(x) = a(x^2 + 3x - 7x - 21) = a(x^2 - 4x - 21)$ So, the quadratic function becomes: $f(x) = ax^2 - 4ax - 21a$

2. Identifying $a$ and $b$: From this expanded form, we can identify that:

   • $b = -4a$
   • We are asked to find $a + b$, which is: $a + b = a - 4a = -3a$

3. Finding $a + b$: Since $a$ is an integer greater than 1, the smallest possible value of $a$ is 2. Let's calculate $a + b$ for $a = 2$: $a + b = -3a = -3(2) = -6$

Thus, the value of $a + b$ is $\boxed{-6}$.

Would you like more details or have any further questions?

Here are 5 related questions to consider:

1. How do you factor a quadratic equation given its roots?
2. What are other methods to find the vertex of a quadratic function?
3. How does the discriminant of a quadratic equation relate to its graph?
4. Can the quadratic function have more than two real roots?
5. How does the value of $a$ in $ax^2 + bx + c$ affect the shape of the parabola?

Tip: When dealing with quadratic functions, factoring is often the easiest method to identify roots and rewrite the equation in standard form.