Can someone check my answers for the following problems.

Simplify. Assume that no denominator equals 0.
1)(7x^2+3x-9)-(-x^2+8x-3)
answer: 6x^2-5x-6

2) 5x^3(7x)^2
answer: 245x^5

3) (2x-3)^2
answer: 4x^2-9

4) Use synthetic division to find (2x^3-5x^2+7x-1)/(x-1)
answer: 3

5)Identify the vertex,axis of symmetry, and direction of opening for y=2(x+3)^2-5
answer:
Axis of symmetry is x= -3/4
vertex -3/4
opens up

6) write y=-4x^2+8x-1 in vertex form
answer: (-4x+4)^2-17

1)(7x^2+3x-9)-(-x^2+8x-3)

answer: 6x^2-5x-6
now wait a minute
7 x^2 - ( -x^2) = 7 x^2 + x^2 = 8 x^2
agree with -5x-6

3) (2x-3)^2
answer: 4x^2-9
NO
(2x-3)(2x-3) = 4x^2-12 x + 9

now if you factor 4x^2 - 9 you get (2x-3)(2x+3)

4) Use synthetic division to find (2x^3-5x^2+7x-1)/(x-1)

answer: 3
Well I get a remainder of 3
2x^2 - 3 x + 4 and remainder of 3

5)Identify the vertex,axis of symmetry, and direction of opening for y=2(x+3)^2-5

answer:
Axis of symmetry is x= -3/4
vertex -3/4
opens up
===============
yes, opens up because as |x| gets big, y gets big +
(x+3) = 0 at x = -3
That is axis of symmetry
when x = -3
y = -5
so vertex at (-3,-5)

6) write y=-4x^2+8x-1 in vertex form

answer: (-4x+4)^2-17

y+1 = -4 x^2 + 8 x
-(y+1)/4 = x^2 - 2x
complete square by adding (-2/2)^2 to both sides
-(y+1)/4 + 1 = x^2 -2 x +1
-(y+1)/4 + 1 = (x-1)^2
(x-1)^2 = -y/4 -1/4 +4/4
(x-1)^2 = (-1/4)(y-3)
vertex (1,+3)

For #1-3 I have to simplify them...and for #1 would it be 8x^2-5x-6 in the book it tells me to use the FOIL method.

and for #1 would it be 8x^2-5x-6

Yes

3) (2x-3)^2
answer: 4x^2-9

NO
(2x-3)(2x-3) = 4x^2-12 x + 9 <--- That is FOIL

To check the answers to these problems, you can follow these steps for each problem:

1) Simplifying:
Expand both sets of parentheses and combine like terms.
(7x^2+3x-9)-(-x^2+8x-3)
= 7x^2 + 3x - 9 + x^2 - 8x + 3
= (7x^2 + x^2) + (3x - 8x) + (-9 + 3)
= 8x^2 - 5x - 6

Compare the simplified expression with the given answer:
Your answer is: 6x^2 - 5x - 6
The correct answer is: 8x^2 - 5x - 6

2) Multiplying:
Apply the exponent rules and simplify the expression.
5x^3(7x)^2 = 5x^3(49x^2) = 245x^5

Compare the simplified expression with the given answer:
Your answer is: 245x^5
The correct answer is: 245x^5

3) Expanding:
Apply the exponent rules to expand the expression.
(2x-3)^2 = (2x-3)(2x-3) = 2x*2x + 2x*(-3) + (-3)*2x + (-3)*(-3)
= 4x^2 - 6x - 6x + 9
= 4x^2 - 12x + 9

Compare the expanded expression with the given answer:
Your answer is: 4x^2 - 9
The correct answer is: 4x^2 - 12x + 9

4) Synthetic division:
Use synthetic division to divide the polynomial (2x^3-5x^2+7x-1) by (x-1).
The dividend is 2x^3 - 5x^2 + 7x - 1, and the divisor is x - 1.

2 -5 7 -1
1 | 2 -5 7 -1
2 -3 4 3

The result is 2x^2 - 3x + 4 with a remainder of 3.

Compare the result of synthetic division with the given answer:
Your answer is: 3
The correct answer is: 3

5) Vertex form and characteristics:
To find the vertex form of y = -4x^2 + 8x - 1, you need to complete the square.

Begin by factoring out the leading coefficient (-4).
y = -4(x^2 - 2x + 1/4) - 1 + 4(-1/4)
y = -4(x - 1/2)^2 - 1 + 1
y = -4(x - 1/2)^2

The vertex form is given by y = a(x - h)^2 + k, where (h, k) represents the vertex.

Axis of symmetry (h) is given by -b/2a. In this case, h = -(-2) / (2*(-4)) = 1/2.
The vertex is (1/2, 0).

Since the leading coefficient is negative, the parabola opens downward (not upward).

Compare the characteristics with the given answer:
Your answer is: Axis of symmetry is x = -3/4, vertex is -3/4, opens up
The correct answer is: Axis of symmetry is x = -3/4, vertex is (1/2, 0), opens downward

6) Writing in vertex form:
To write y = -4x^2 + 8x - 1 in vertex form, complete the square.

Begin by factoring out the leading coefficient (-4).
y = -4(x^2 - 2x) - 1

Next, take half of the coefficient of x (-2), square it (-2)^2 = 4, and add it inside the parentheses. Add the same amount outside the parentheses to balance the equation.
y = -4(x^2 - 2x + 4 - 4) - 1 + 4

Simplify the equation.
y = -4((x - 2)^2 - 4) - 1 + 4
y = -4(x - 2)^2 + 16 - 1 + 4
y = -4(x - 2)^2 + 19

Compare the rewritten equation with the given answer:
Your answer is: (-4x + 4)^2 - 17
The correct answer is: -4(x - 2)^2 + 19

Rechecking your answers in this manner will help ensure accuracy.