a. 225:15+3x(2xX+1)=270

To solve this equation, we will follow the order of operations (PEMDAS/BODMAS).

First, let's simplify the expression inside the parentheses:
2xX + 1

Next, we multiply 3 by the simplified expression:
3(2xX + 1) = 6xX + 3

Now, let's substitute the simplified expression back into the equation:
225:15 + 6xX + 3 = 270

Next, let's combine like terms:
240 + 6xX = 270

Now, let's isolate the variable by subtracting 240 from both sides:
6xX = 30

Finally, let's solve for X by dividing both sides by 6:
X = 5

Therefore, the solution to the equation is X = 5.

b. 4x(2xX+1)^3 = 108

To solve this equation, we will follow the order of operations (PEMDAS/BODMAS).

First, let's simplify the expression inside the parentheses:
2xX + 1

Next, we substitute the simplified expression back into the equation:
4x(2xX + 1)^3 = 108

Now, let's simplify the exponent:
(2xX + 1)^3 = (2xX + 1)(2xX + 1)(2xX + 1) = (2xX)^3 + 3(2xX)^2(1) + 3(2xX)(1)^2 + (1)^3 = 8x^3 + 12x^2 + 6x + 1

Now, let's substitute the simplified expression back into the equation:
4x(8x^3 + 12x^2 + 6x + 1) = 108

Next, let's distribute 4x to each term inside the parentheses:
32x^4 + 48x^3 + 24x^2 + 4x = 108

Now, let's combine like terms:
32x^4 + 48x^3 + 24x^2 + 4x - 108 = 0

Unfortunately, this equation cannot be solved algebraically. You would need to use numerical methods or a graphing calculator to find an approximate solution.

c. 15x2^3 - (5x - 3) = 108

To solve this equation, we will follow the order of operations (PEMDAS/BODMAS).

First, let's simplify the exponent:
2^3 = 2 * 2 * 2 = 8

Now, let's substitute the simplified exponent back into the equation:
15x8 - (5x - 3) = 108

Next, let's distribute the negative sign to the terms inside the parentheses:
120x - 5x + 3 = 108

Now, let's combine like terms:
115x + 3 = 108

Next, let's isolate the variable by subtracting 3 from both sides:
115x = 105

Finally, let's solve for X by dividing both sides by 115:
X = 105/115

Therefore, the solution to the equation is X = 105/115.

d. 19x3^2 - 9x(7xX - 2) = 0

To solve this equation, we will follow the order of operations (PEMDAS/BODMAS).

First, let's simplify the exponent:
3^2 = 3 * 3 = 9

Now, let's substitute the simplified exponent back into the equation:
19x9 - 9x(7xX - 2) = 0

Next, let's distribute 9x to the terms inside the parentheses:
171x - 63x^2 + 18x = 0

Now, let's combine like terms:
171x - 63x^2 + 18x = 0

Next, let's rearrange the equation to make it easier to solve:
-63x^2 + 189x = 0

Now, let's factor out the common factor of x:
x(-63x + 189) = 0

Now, let's set each factor equal to zero and solve for x:
x = 0 or -63x + 189 = 0

For the second equation, let's isolate x by subtracting 189 from both sides:
-63x = -189

Finally, let's solve for x by dividing both sides by -63:
x = -189/-63

Therefore, the solutions to the equation are x = 0 and x = 3.