### Solution:
Let's start by writing the given polynomial expression in standard form and then evaluate it for [tex]\( x = 10 \)[/tex].
#### a. Writing the Expression in Standard Form and Stating the Degree
The given polynomial expression is:
[tex]\[ 3 - 5x^2 + 2 + 8x + x^3 + 7x^2 - 5 \][/tex]
To simplify this expression, we first combine the constant terms and like terms.
#### Step-by-Step Simplification:
1. Combine the constant terms:
[tex]\[ 3 + 2 - 5 = 0 \][/tex]
2. Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ -5x^2 + 7x^2 = 2x^2 \][/tex]
3. Combine any remaining terms:
[tex]\[ 8x \][/tex] and [tex]\[ x^3 \][/tex] remain as they are since they do not have like terms.
Putting all these together, the simplified polynomial expression is:
[tex]\[ x^3 + 2x^2 + 8x - 5 \][/tex]
Thus, the polynomial in standard form is:
[tex]\[ x^3 + 2x^2 + 8x - 5 \][/tex]
The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex]. In this polynomial, the highest power is 3. Therefore, the degree of the polynomial is 3.
#### b. Evaluating the Expression for [tex]\( x = 10 \)[/tex]
Now, we evaluate the polynomial [tex]\( x^3 + 2x^2 + 8x - 5 \)[/tex] for [tex]\( x = 10 \)[/tex].
Substitute [tex]\( x = 10 \)[/tex] into the polynomial:
[tex]\[ (10)^3 + 2(10)^2 + 8(10) - 5 \][/tex]
Calculate each term:
1. [tex]\( (10)^3 = 1000 \)[/tex]
2. [tex]\( 2(10)^2 = 2(100) = 200 \)[/tex]
3. [tex]\( 8(10) = 80 \)[/tex]
4. The constant term remains as [tex]\(-5\)[/tex].
Now, sum these values:
[tex]\[ 1000 + 200 + 80 - 5 \][/tex]
The result is:
[tex]\[ 1275 \][/tex]
Therefore, the evaluated expression for [tex]\( x = 10 \)[/tex] is [tex]\( 1275 \)[/tex].
### Summary:
a. The polynomial in standard form is:
[tex]\[ x^3 + 2x^2 + 8x - 5 \][/tex]
And its degree is: 3
b. The value of the polynomial for [tex]\( x = 10 \)[/tex] is: 1275
