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We are tasked with factoring the quadratic expression \( -3x^2 + 45x - 78 \).

### Step 1: Factor out the greatest common factor (GCF)
The GCF of the coefficients \(-3\), \(45\), and \(-78\) is \(-3\). So, we can factor \(-3\) from the entire expression:

\[
-3(x^2 - 15x + 26)
\]

### Step 2: Factor the quadratic inside the parentheses
Now, we need to factor \(x^2 - 15x + 26\). To do this, we need to check if it factors into two binomials of the form \((x - p)(x - q)\), but we notice that it doesn't factor nicely.

We can check the discriminant (\(\Delta\)) to determine whether the quadratic has real roots:

\[
\Delta = b^2 - 4ac = (-15)^2 - 4(1)(26) = 225 - 104 = 121
\]

Since \(\Delta = 121\) is a perfect square, the quadratic can be factored using the quadratic formula:
\[
x = \frac{-(-15) \pm \sqrt{121}}{2(1)} = \frac{15 \pm 11}{2}
\]
So the roots are:
\[
x = \frac{15 + 11}{2} = 13 \quad \text{and} \quad x = \frac{15 - 11}{2} = 2
\]

### Step 3: Write the factored form
Thus, the factorization of \(x^2 - 15x + 26\) is:
\[
(x - 13)(x - 2)
\]

### Final Answer:
Now, we include the \(-3\) factor back:
\[
-3(x - 13)(x - 2)
\]

This is the fully factored form of the quadratic expression.

Let me know if you want more details or have any questions!

Here are 5 related questions:
1. How can we determine whether a quadratic trinomial is factorable?
2. What happens when the discriminant is negative or zero?
3. How do you solve a quadratic using the quadratic formula?
4. Can you explain how to complete the square for quadratics?
5. How do you graph a quadratic function from its factored form?

**Tip:** Always check the discriminant to confirm whether a quadratic is factorable into real numbers.

Factor the quadratic expression -3x^2 + 45x - 78.

This page explains how to factor the quadratic expression -3x^2 + 45x - 78. The solution uses the greatest common factor and quadratic formula, making it suitable for Grades 9-11 students learning about factoring and solving quadratic equations.

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