im not sure how to solve this problem by using the box method. can you please show step by step by using the box method please and thank you

It's not clearly stated what you are looking to solve for in this problem. You're probably either being asked to expand the polynomial or to factor the part of the problem that's not completely factored (that being the trinomial since the binomial can't be factored any further.

If it asks you to expand the polynomial, then you need to use the distributive property to multiply the binomial by the trinomial:

(x + 3)(x^{2} + 4x - 5)

= (x·x^{2} + x·4x - x·5) + (3·x^{2} + 3·4x - 3·5)

= x^{3} + **4x**^{2}
*- 5x* **+ 3x**^{2} *+ 12x* - 15

= x^{3} + 7x^{2} + 7x - 15

If it asks you to factor completely, this would indicate that you need to factor the 2nd part of the problem (the trinomial) since the first part (the binomial) can't be factored any further:

(x + 3)(x^{2} + 4x - 5)

Factor: x^{2} + 4x - 5

Using the box method, draw a square box and draw a line down the middle in both directions so as to have 4 boxes inside the square. Put the first term of the trinomial (x^{2}) in the box in the upper left hand corner and the last term of the trinomial (-5) in the box in the lower right hand corner.

The terms that go in the remaining two boxes are determined as follows:

Find the value of product of the coefficient of the first term and the last term (the constant term). That is, the product of the coefficient of x^{2} which is 1 and the last term which is -5.....

.... 1 · -5 = **-5**

Find the factor pairs of this value...

... factor pairs of -5: -1 and 5 or 1 and -5

Choose the pair whose sum is the coefficient of the middle term in the trinomial. That is, choose the pair that add up to equal +4.....

... -1 + 5 = 4 , 1 + -5 = -4

The factor pair that add up to positive 4 is the first pair, that being -1 and 5. Place -1x in one of the boxes and 5x in the other box.

Now, find the greatest common factor among each set of 2 terms vertically and horizontally. The 2 terms factored from the vertical sets make up one binomial and the 2 terms factored from the horizontal sets make up the other binomial.

**x -1**

_________________

**x** ¦ x^{2} ¦ -1x

_________________

**5** ¦ 5x ¦ -5

_________________

So, x^{2} + 4x - 5 = (x - 1)(x + 5)

Therefore,

(x + 3)(x^{2} + 4x - 5) = (x + 3)(x - 1)(x + 5)