We subtract polynomials just like we would anything else, but we must pay attention to the sign after using the distributive property on the sign that separates the two polynomials in subtraction problems. Then, of course, only like terms can combine or subtract from each other.

$\displaystyle (6x^4+5x^3+12x^2+2x+5)-(6x^4+5x^3+11x^2+2x+5)$

Start by distributing the negative sign into the second polynomial:

$\displaystyle 6x^4+5x^3+12x^2+2x+5-6x^4-5x^3-11x^2-2x-5)$

Now combine or subtract like terms based on the sign between them (same colored terms cancel each other out):

$\displaystyle {\color{Red} 6x^4}+{\color{Green} 5x^3}+12x^2+{\color{Blue} 2x}+{\color{Purple} 5}-{\color{Red} 6x^4}-{\color{Green} 5x^3}-11x^2-{\color{Blue} 2x}-{\color{Purple} 5})$

Left over: $\displaystyle 12x^2-11x^2=x^2$

$\displaystyle (4x^3+6xy+2y-37)-(6x^2+6xy+y+14)$

Use distribution to simplify the signs of the polynomials.

$\displaystyle 4x^3+6{\color{Magenta} xy}+{2\color{Cyan}y}-{{\color{DarkOrange} 37}}-6x^2-6{\color{Magenta} xy}-{\color{Cyan} y}-{\color{DarkOrange} 14}$

Combine like terms while following coefficient rules.

$\displaystyle 4x^3+(6-6){{\color{Magenta} xy}}+{(2-1){\color{Cyan} y}}-6x^2-({\color{DarkOrange} {37}-{14}})$

The correct expanded and simplified answer will be

$\displaystyle 4x^3-6x^2+y-51$.

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle (3x^2 - y) - (x^2 + 2)$

$\displaystyle 3x^2 - y - x^2 - 2$

$\displaystyle 2x^2 - y - 2$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle (3x^2 + 2x + 1) - (2x^2 - 2x + 5)$

$\displaystyle 3x^2 + 2x + 1 - 2x^2 + 2x - 5$

$\displaystyle x^2 + 4x - 4$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle y(x +1) - y(x^2 + 1)$

$\displaystyle yx + y - yx^2 - y$

$\displaystyle -yx^2 + yx$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle (2ab + 4) - x(ab + 3)$

$\displaystyle 2ab + 4 -xab -3$

$\displaystyle 2ab -xab +1$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle 2x^2 - y - (4x + y)$

$\displaystyle 2x^@ - y -4x - y$

$\displaystyle 2x^2 - 4x - 2y$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle 2(x^2 + y^2) - 2(x - y)$

$\displaystyle 2x^2 + 2y^2 - 2x + 2y$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle a(a + b) - b(a+b)$

$\displaystyle a^2 + ab - ab - b^2$

$\displaystyle a^2 -b^2$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle a(x + y) - a(x-y)$

$\displaystyle ax + ay - ax + ay$

$\displaystyle 2ay$