Let's evaluate the polynomial expression:
[tex]\[
5(2x^2 - 3xy + y^2 - 3) - 2x(x + 7y - 1) - 3y^2
\][/tex]
Step-by-step, plug in [tex]\(x = -3\)[/tex] and [tex]\(y = 4\)[/tex]:
First, evaluate the expression inside the first parentheses:
[tex]\[
2x^2 - 3xy + y^2 - 3
\][/tex]
Substitute [tex]\(x = -3\)[/tex] and [tex]\(y = 4\)[/tex]:
[tex]\[
2(-3)^2 - 3(-3)(4) + 4^2 - 3
\][/tex]
Calculate each term:
[tex]\[
2(9) - 3(-12) + 16 - 3 = 18 + 36 + 16 - 3
\][/tex]
Combine the results:
[tex]\[
18 + 36 + 16 - 3 = 67
\][/tex]
Then, multiply by 5:
[tex]\[
5 \cdot 67 = 335
\][/tex]
Now, evaluate the second term:
[tex]\[
-2x(x + 7y - 1)
\][/tex]
Substitute [tex]\(x = -3\)[/tex] and [tex]\(y = 4\)[/tex]:
[tex]\[
-2(-3)((-3) + 7(4) - 1) = -2(-3)(-3 + 28 - 1) = -2(-3)(24)
\][/tex]
Simplify the multiplication:
[tex]\[
-2(-3) \cdot 24 = 6 \cdot 24 = 144
\][/tex]
Finally, evaluate the last term:
[tex]\[
-3y^2
\][/tex]
Substitute [tex]\(y = 4\)[/tex]:
[tex]\[
-3(4^2) = -3 \cdot 16 = -48
\][/tex]
Combine all the parts:
[tex]\[
335 + 144 - 48
\][/tex]
Add the results step-by-step:
[tex]\[
335 + 144 = 479
\][/tex]
Then subtract:
[tex]\[
479 - 48 = 431
\][/tex]
Therefore, the value of the polynomial expression for [tex]\(x = -3\)[/tex] and [tex]\(y = 4\)[/tex] is:
[tex]\[
\boxed{431}
\][/tex]
