Sure, let's solve the expression [tex]\((m+n+p)^2 - (m-n+p)^2\)[/tex] step-by-step.
### Step 1: Understanding the Expression
We need to evaluate the difference of the squares of two expressions: [tex]\(a = (m+n+p)\)[/tex] and [tex]\(b = (m-n+p)\)[/tex].
### Step 2: Using the Difference of Squares Formula
Recall that the difference of squares formula is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, our [tex]\(a\)[/tex] is [tex]\((m+n+p)\)[/tex] and our [tex]\(b\)[/tex] is [tex]\((m-n+p)\)[/tex].
### Step 3: Calculate [tex]\(a - b\)[/tex]
Let's substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into [tex]\(a - b\)[/tex]:
[tex]\[
a - b = (m+n+p) - (m-n+p)
\][/tex]
Simplifying this:
[tex]\[
a - b = m + n + p - m + n - p
\][/tex]
The [tex]\(m\)[/tex] and [tex]\(-m\)[/tex] cancel each other out, and the [tex]\(p\)[/tex] and [tex]\(-p\)[/tex] cancel each other out:
[tex]\[
a - b = 2n
\][/tex]
### Step 4: Calculate [tex]\(a + b\)[/tex]
Now let’s substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into [tex]\(a + b\)[/tex]:
[tex]\[
a + b = (m+n+p) + (m-n+p)
\][/tex]
Simplifying this:
[tex]\[
a + b = m + n + p + m - n + p
\][/tex]
The [tex]\(+n\)[/tex] and the [tex]\(-n\)[/tex] cancel each other out:
[tex]\[
a + b = 2m + 2p
\][/tex]
### Step 5: Multiply [tex]\(a - b\)[/tex] and [tex]\(a + b\)[/tex]
From the previous steps, we have:
[tex]\[
a - b = 2n
\][/tex]
[tex]\[
a + b = 2m + 2p
\][/tex]
Now, multiply these two results:
[tex]\[
(a - b)(a + b) = (2n)(2m + 2p)
\][/tex]
Distribute [tex]\(2n\)[/tex] across [tex]\(2m + 2p\)[/tex]:
[tex]\[
(2n)(2m) + (2n)(2p) = 4mn + 4np
\][/tex]
### Step 6: Final Answer
Thus, [tex]\((m+n+p)^2 - (m-n+p)^2\)[/tex] evaluates to:
[tex]\[
4mn + 4np
\][/tex]
Considering the given specific values, [tex]\(m = 1\)[/tex], [tex]\(n = 2\)[/tex], and [tex]\(p = 3\)[/tex], the numerical result of the expression is:
[tex]\[
32
\][/tex]
So, the solution is [tex]\(4mn + 4np = 32\)[/tex].
