Answer :

To add the polynomials

[tex]$$4x^4 - 9x^3 - 5x \quad \text{and} \quad 8x^4 - 10x^3 + 3,$$[/tex]

we follow these steps:

1. Identify Like Terms:

- The terms with [tex]$x^4$[/tex] are [tex]$4x^4$[/tex] and [tex]$8x^4$[/tex].
- The terms with [tex]$x^3$[/tex] are [tex]$-9x^3$[/tex] and [tex]$-10x^3$[/tex].
- The term with [tex]$x$[/tex] is [tex]$-5x$[/tex] (from the first polynomial, with no corresponding [tex]$x$[/tex] term in the second).
- The constant term is [tex]$3$[/tex] (from the second polynomial).

2. Combine the Like Terms:

- For the [tex]$x^4$[/tex] terms:

[tex]$$
4x^4 + 8x^4 = 12x^4.
$$[/tex]

- For the [tex]$x^3$[/tex] terms:

[tex]$$
-9x^3 - 10x^3 = -19x^3.
$$[/tex]

- The [tex]$x$[/tex] term remains as:

[tex]$$
-5x.
$$[/tex]

- The constant term remains as:

[tex]$$
3.
$$[/tex]

3. Write the Resulting Polynomial:

Combine the results from the previous step to obtain:

[tex]$$
12x^4 - 19x^3 - 5x + 3.
$$[/tex]

Thus, the sum of the polynomials is

[tex]$$\boxed{12x^4 - 19x^3 - 5x + 3.}$$[/tex]

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