How do you simplify # (2x^2 - 5x - 1)(x^2 + x - 3) #?

Question

Eva Abramson · Accepted Answer

#(2x^2-5x+1)xx(x^2+x-3) = 2x^4-3x^3-12x^2+14x+3#

#{: ( xx," |",2x^2, -5x, -1), ("-----", ,"-----","-----","-----"), (x^2," |",color(orange)(2x^4),color(blue)(-5x^3),color(red)(-x^2)), (+x," |",color(blue)(2x^3),color(red)(-5x^2),color(green)(-x)), (-3," |",color(red)(-6x^2),color(green)(+15x),color(cyan)(+3)), ("-----","-----","-----","-----","-----"), (color(orange)(2x^4),color(blue)(-3x^3),color(red)(-12x^2),color(green)(+14x),color(cyan)(+3)) :}#

Abigail Allen · Answer

undefined To simplify the expression $ (2x^2 - 5x - 1)(x^2 + x - 3) $:

1. First, distribute each term in the first polynomial by each term in the second polynomial.
2. Then, combine like terms.

Following these steps, we get:

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

$$ = 2x^4 + 2x^3 - 6x^2 - 5x^3 - 5x^2 + 15x - x^2 - x - 3 $$

$$ = 2x^4 + (2x^3 - 5x^3) + (-6x^2 - 5x^2 - x^2) + (15x - x) - 3 $$

$$ = 2x^4 - 3x^3 - 12x^2 + 14x - 3 $$

Therefore, the simplified expression is $ 2x^4 - 3x^3 - 12x^2 + 14x - 3 $.