To find the value of k, we need to divide the polynomial x^3 - 3x^2 + 5x - 3 by the binomial x^2 - k and determine the remainder. Given that the remainder is 7x, we can set up the division as follows:

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x^2 - k | x^3 - 3x^2 + 5x - 3

We will perform polynomial long division to find the quotient and remainder.

Step 1: Divide the first term of the dividend by the first term of the divisor.

x^3 / x^2 = x

Step 2: Multiply the entire divisor by the quotient obtained in Step 1.

(x^2 - k) * x = x^3 - kx

Step 3: Subtract the result obtained in Step 2 from the dividend.

(x^3 - 3x^2 + 5x - 3) - (x^3 - kx) = -3x^2 + (5 - k)x - 3

Step 4: Bring down the next term from the dividend.

-3x^2 + (5 - k)x - 3

Step 5: Repeat the process by dividing the first term of the new expression by the first term of the divisor.

-3x^2 / x^2 = -3

Step 6: Multiply the entire divisor by the quotient obtained in Step 5.

(x^2 - k) * -3 = -3x^2 + 3k

Step 7: Subtract the result obtained in Step 6 from the new expression.

(-3x^2 + (5 - k)x - 3) - (-3x^2 + 3k) = (5 - k - 3k)x - 3 + 3k = (5 - 4k)x - 3 + 3k

Step 8: Bring down the next term from the dividend.

(5 - 4k)x - 3 + 3k

Step 9: Repeat the process by dividing the first term of the new expression by the first term of the divisor.

(5 - 4k)x / x^2 = (5 - 4k) / x

Step 10: Multiply the entire divisor by the quotient obtained in Step 9.

(x^2 - k) * ((5 - 4k) / x) = (5 - 4k)x - (k(5 - 4k)) / x

Step 11: Subtract the result obtained in Step 10 from the new expression.

((5 - 4k)x - 3 + 3k) - ((5 - 4k)x - (k(5 - 4k)) / x) = - (k(5 - 4k)) / x + 3k - 3

Step 12: The remainder is 7x, so we can set up an equation with the last expression equal to 7x.

- (k(5 -