Factor the trinomial completely using the trial and error method.
negative 2 x squared minus 17 x plus 55

To factor the trinomial \(-2x^2 - 17x + 55\) using the trial and error method, follow these steps:

### 1. Identify the coefficients:
The given trinomial is \(-2x^2 - 17x + 55\).

- The coefficient of \(x^2\) (leading coefficient) is \(a = -2\).
- The coefficient of \(x\) is \(b = -17\).
- The constant term is \(c = 55\).

### 2. Look for factor pairs of \(a \times c\):
First, multiply the leading coefficient by the constant term:
\[
a \times c = (-2) \times 55 = -110
\]
Now, find pairs of factors of \(-110\) that will add up to \(b = -17\).

Possible factor pairs of \(-110\) are:
\[
(-1, 110),\ (-2, 55),\ (-5, 22),\ (-10, 11)
\]
Now, check which pair sums to \(-17\):
\[
-10 + 11 = 1 \quad (\text{not a match})
\]
\[
-5 + 22 = 17 \quad (\text{not a match})
\]
\[
10 - 11 = -1 \quad (\text{not a match})
\]
\[
-10 + 11 = 1 \quad (\text{not a match})
\]
\[
5 - 22 = -17 \quad (\text{this matches!})
\]

So, the correct factors are \(-22\) and \(5\).

### 3. Rewrite the middle term using the found factors:
\[
-2x^2 - 22x + 5x + 55
\]

### 4. Factor by grouping:
Group the terms:
\[
(-2x^2 - 22x) + (5x + 55)
\]
Factor out the greatest common factor from each group:
\[
-2x(x + 11) + 5(x + 11)
\]
Notice that \((x + 11)\) is a common factor:
\[
(x + 11)(-2x + 5)
\]

### 5. Final Answer:
The completely factored form of the trinomial is:
\[
(x + 11)(-2x + 5)
\]

Would you like more details or have any questions?

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**Related Questions:**
1. How would you factor a trinomial with a positive leading coefficient using the trial and error method?
2. What happens if the factor pairs do not match the middle term's coefficient?
3. Can you use the quadratic formula to double-check the roots of the factored form?
4. How do you factor trinomials when the middle term is positive?
5. What is the difference between factoring by grouping and the trial and error method?

**Tip:** Always check your factors by expanding them to ensure they multiply back to the original trinomial.

Learn how to factor the trinomial -2x^2 - 17x + 55 completely using the trial and error method with detailed steps and explanations. This problem involves factoring quadratic equations with a negative leading coefficient. Suitable for high school students studying algebra.

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