Answer :

To solve the expression [tex]\(\left(12x^3 + 25x^2 - 15\right) - \left(-2x^3 + 18x^2 - 3x\right)\)[/tex], follow these steps:

1. Distribute the negative sign: When we subtract polynomials, it's like adding the opposite of the second polynomial. So, begin by distributing the negative sign through the second polynomial:

[tex]\[
-\left(-2x^3 + 18x^2 - 3x\right) = 2x^3 - 18x^2 + 3x
\][/tex]

2. Combine the polynomials: Now, add the expressions from the first polynomial to the modified second polynomial:

[tex]\[
\left(12x^3 + 25x^2 - 15\right) + \left(2x^3 - 18x^2 + 3x\right)
\][/tex]

3. Combine like terms: Add coefficients of terms with the same degree:

- For [tex]\(x^3\)[/tex] terms: [tex]\(12x^3 + 2x^3 = 14x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(25x^2 - 18x^2 = 7x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(0 + 3x = 3x\)[/tex] (note: there is no [tex]\(x\)[/tex] term in the first polynomial, so we assume it as 0)
- For the constant terms: [tex]\(-15 + 0 = -15\)[/tex] (note: there is no constant in the modified second polynomial, so we assume it as 0)

4. Write the final result: After combining the like terms, the resulting polynomial is:

[tex]\[
14x^3 + 7x^2 + 3x - 15
\][/tex]

This simplified polynomial is the solution to the given expression.

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