To find \( m(x+y+z) \) for the function \( m(x) = x^2 + 3x - 5 \), we need to substitute \( x \) with \( x + y + z \) in the expression.

Given:
\[ m(x) = x^2 + 3x - 5 \]

We need to find \( m(x+y+z) \):
\[ m(x+y+z) = (x+y+z)^2 + 3(x+y+z) - 5 \]

First, expand \( (x+y+z)^2 \):
\[ (x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz \]

Next, expand \( 3(x+y+z) \):
\[ 3(x+y+z) = 3x + 3y + 3z \]

Now, combine these expansions into the original function:
\[ m(x+y+z) = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz + 3x + 3y + 3z - 5 \]

So, the expression for \( m(x+y+z) \) is:
\[ m(x+y+z) = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz + 3x + 3y + 3z - 5 \]

Would you like more details or have any questions about this process?

Here are 8 relative questions expanding on this information:

1. How would you find \( m(x-y+z) \) for the function \( m(x) = x^2 + 3x - 5 \)?
2. What is the value of \( m(1+2+3) \)?
3. How do you simplify the expression \( (x+y+z)^2 \)?
4. How do you find the roots of the quadratic function \( m(x) = x^2 + 3x - 5 \)?
5. What is the vertex of the parabola represented by \( m(x) = x^2 + 3x - 5 \)?
6. How do you graph the function \( m(x) = x^2 + 3x - 5 \)?
7. How would you solve for \( x \) when \( m(x) = 0 \)?
8. What are the transformations applied to the function \( m(x) = x^2 \) to get \( m(x) = x^2 + 3x - 5 \)?

**Tip:** When dealing with polynomial expressions, make sure to expand and combine like terms carefully to avoid mistakes.

Learn how to find m(x+y+z) for the function m(x) = x^2 + 3x - 5 with detailed step-by-step solution. This problem covers quadratic functions, polynomial expansion, and algebraic manipulation. Suitable for advanced high school students.

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