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------------------------------------------------ Divide the following: [tex]{10x^{4} +19x^{3}-32x^{2}-4x-3}[/tex] by [tex]5x^{2}+3x-1[/tex]

When dividing [tex]$10x^4 + 19x^3 - 32x^2 - 4x - 3$[/tex] by [tex]$5x^2 + 3x - 1$[/tex], the quotient is [tex]$2x^2 + 2x - 1$[/tex] with no remainder.

To divide [tex]$10x^4 + 19x^3 - 32x^2 - 4x - 3$[/tex] by [tex]$5x^2 + 3x - 1$[/tex], we perform polynomial long division. Here's how:

1. Divide the highest-degree term of the dividend by the highest-degree term of the divisor to get the first term of the quotient.

- [tex]$10x^4 \div 5x^2 = 2x^2$[/tex].

2. Multiply the entire divisor by the first term of the quotient and subtract it from the dividend.

- [tex]$(2x^2)(5x^2 + 3x - 1) = 10x^4 + 6x^3 - 2x^2$[/tex].

- Subtracting from the original dividend gives [tex]$19x^3 - 32x^2 - 4x - 3 - (10x^4 + 6x^3 - 2x^2) = 13x^3 - 30x^2 - 4x - 3$.[/tex]

3. Repeat the process with the new dividend.

- Divide [tex]$13x^3$ by $5x^2$[/tex] to get 2x.

- Multiply and subtract to get [tex]$13x^3 - 10x^2 - 6x$[/tex].

4. Repeat until the degree of the remainder is less than the degree of the divisor.

The quotient is [tex]$2x^2 + 2x - 1$[/tex] with a remainder of 0.