Question

Multiply the polynomials (x+4)(x^2-5x+3)

Answer 1

Answer:

$\\ x^{3} - x^{2} - 17x + 12$

Step-by-step explanation:

Write the polynomials one above the other to multiply them:

$\\ x^{2} -5x + 3$

         $\\ x + 4$

First Step

Multiply each element of $\\ x^{2} -5x + 3$ by 4:

The result is: $\\ 4x^{2} - 20x + 12$ [1]

As it can be seen, only the coefficients are multiplied by 4, and when the operation involves multiplication with coefficients with different operators (+ or -), the following rules are crucial:  

$\\ + * + = +$

$\\ - * - = +$

$\\ + * - = -$

$\\ - * + = -$

That is why 4 times -5 = -20,  4 times 1 = 4, and so 4 times 3 = 12.

Second Step

Multiply each element of $\\ x^{2} -5x + 3$ by x:

The same rules apply here, but including the addition of powers or exponents.

The result is: $\\ x^{3} - 5x^{2} + 3x$ [2].

$\\ x * x^{2} = x^{3}$ or $\\ x * x^{2} = x^{1+2} =x^{3}$

$\\ x * (-5x)=-5x^{2}$

$\\ x*3 = 3x$

Third Step

Sum all similar terms of the previous results [1] and [2].

The result is: $\\ x^{3}-x^{2}-17x +12$, because

         $\\ 4x^{2} - 20x + 12$

$\\ x^{3} - 5x^{2} + 3x$

__________________________

$\\ x^{3} + (4-5)x^{2} + (-20+3)x+12$

$\\ x^{3} + (-1)x^{2} + (-17)x+12$

$\\ x^{3} - x^{2} -17x+12$.

Answer 2

(4x2 – 4x – 7)(x + 3) 
  =  (4x2 – 4x – 7)(x) + (4x2 – 4x – 7)(3) 
  =  4x2(x) – 4x(x) – 7(x) + 4x2(3) – 4x(3) – 7(3) 
  =  4x3 – 4x2 – 7x + 12x2 – 12x – 21 
  =  4x3 – 4x2 + 12x2 – 7x – 12x – 21 
  =  4x3 + 8x2 – 19x – 21