To subtract the polynomials [tex]\( (10x + 5) - (-12x + 7) \)[/tex], we will follow a step-by-step process.
1. Rewrite the subtraction as adding the opposite sign of the second polynomial:
We start with:
[tex]\[
(10x + 5) - (-12x + 7)
\][/tex]
Subtracting a polynomial is the same as adding its opposite, therefore:
[tex]\[
(10x + 5) + (-1) \cdot (-12x + 7)
\][/tex]
2. Distribute the [tex]\((-1)\)[/tex] through the second polynomial:
When we multiply [tex]\(-1\)[/tex] by each term in the polynomial [tex]\(-12x + 7\)[/tex], we get:
[tex]\[
(-1) \cdot (-12x) + (-1) \cdot 7 \Rightarrow 12x - 7
\][/tex]
So the expression becomes:
[tex]\[
(10x + 5) + (12x - 7)
\][/tex]
3. Combine the like terms:
- Combine the [tex]\(x\)[/tex]-terms: [tex]\(10x + 12x = 22x\)[/tex]
- Combine the constant terms: [tex]\(5 - 7 = -2\)[/tex]
Therefore, our resulting polynomial is:
[tex]\[
22x - 2
\][/tex]
So, the result of subtracting the polynomial [tex]\((-12x + 7)\)[/tex] from [tex]\((10x + 5)\)[/tex] is:
[tex]\[
\boxed{22x - 2}
\][/tex]
