The trinomial (x^2 - 3 x - 4) is a factor of the function g(x) = 2 x^2 (x - 6) + mx + n, where m and n are integers. The value of (m+n) equals:

Given that \((x^2 - 3x - 4)\) is a factor of \(g(x) = 2x^2(x - 6) + mx + n\), we can start by rewriting \(g(x)\):

\[g(x) = 2x^2(x - 6) + mx + n\]

First, expand the expression:

\[g(x) = 2x^3 - 12x^2 + mx + n\]

Since \((x^2 - 3x - 4)\) is a factor of \(g(x)\), we can write \(g(x)\) as:

\[g(x) = (x^2 - 3x - 4)(2x + k)\]

We need to find the value of \(k\) that satisfies this equation. Let’s expand the right-hand side:

\[(x^2 - 3x - 4)(2x + k) = 2x^3 + kx^2 - 6x^2 - 3kx - 8x - 4k\]

Combining like terms, we get:

\[2x^3 + (k - 6)x^2 + (-3k - 8)x - 4k\]

Comparing coefficients with \(2x^3 - 12x^2 + mx + n\):

\[
\begin{aligned}
    &2x^3 + (k - 6)x^2 + (-3k - 8)x - 4k = 2x^3 - 12x^2 + mx + n \\
    &k - 6 = -12 \\
    &-3k - 8 = m \\
    &-4k = n
\end{aligned}
\]

From \(k - 6 = -12\):

\[k = -6\]

Now, substitute \(k = -6\) into the other equations:

\[
\begin{aligned}
    &-3(-6) - 8 = m \\
    &m = 18 - 8 = 10 \\
    &-4(-6) = n \\
    &n = 24
\end{aligned}
\]

Thus, \(m = 10\) and \(n = 24\). Therefore:

\[m + n = 10 + 24 = 34\]

The value of \(m+n\) is \(34\).

Would you like more details or have any questions? Here are some questions you might ask next:

1. How do you factor polynomials in general?
2. Can you explain polynomial division?
3. What are the applications of factoring in solving equations?
4. How do you determine the roots of a polynomial?
5. What are other methods to factor trinomials?

**Tip:** When factoring polynomials, always check if the polynomial can be written as a product of simpler polynomials by comparing coefficients.

Explore how to find the sum of coefficients m + n in the function g(x) = 2x^2(x - 6) + mx + n, where (x^2 - 3x - 4) is a factor. Learn step-by-step using polynomial factorization and coefficient comparison methods. Suitable for high school students.

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