To simplify the given expression:
[tex]$
\left(8 x^2-3 x+\frac{1}{3}\right)-\left(2 x^2-8 x+\frac{3}{5}\right)
$[/tex]
we can start by distributing the subtraction through the second expression:
[tex]$
8 x^2 - 3 x + \frac{1}{3} - (2 x^2 - 8 x + \frac{3}{5})
$[/tex]
This simplifies to:
[tex]$
8 x^2 - 3 x + \frac{1}{3} - 2 x^2 + 8 x - \frac{3}{5}
$[/tex]
Next, we combine the like terms for [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant terms separately.
#### Step 1: Combine [tex]\(x^2\)[/tex] terms
[tex]$
8 x^2 - 2 x^2 = 6 x^2
$[/tex]
#### Step 2: Combine [tex]\(x\)[/tex] terms
[tex]$
-3 x + 8 x = 5 x
$[/tex]
#### Step 3: Combine the constant terms
[tex]$
\frac{1}{3} - \frac{3}{5}
$[/tex]
To combine these, we need a common denominator, which is 15:
[tex]$
\frac{1}{3} = \frac{5}{15}
$[/tex]
[tex]$
\frac{3}{5} = \frac{9}{15}
$[/tex]
Thus,
[tex]$
\frac{5}{15} - \frac{9}{15} = -\frac{4}{15}
$[/tex]
Putting it all together, the simplified expression is:
[tex]$
6 x^2 + 5 x - \frac{4}{15}
$[/tex]
Therefore, the correct answer is:
[tex]$
6 x^2 + 5 x - \frac{4}{15}
$[/tex]
So, the correct choice is:
[tex]$
6 x^2 + 5 x - \frac{4}{15}
$[/tex]
