Factorise: 10x² - 7ax + (a + 2) (a - 5)
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  (5x -a +5)(2x -a -2)

Step-by-step explanation:

We assume you want to factor this to two linear factors in x and a. A graphing calculator can help immensely (see attached) by showing you the lines you get by setting each factor to zero.

Those lines can be found by considering the x- and a- intercepts.

x-intercepts

Set a=0 and solve for x.

  10x² -10 = 0

  (x -1)(x +1) = 0 . . . . divide by 10; factor the difference of squares

  x = 1 or -1 . . . . . . . . . . zero-product rule

a-intercepts

Set x=0 and solve for a.

  (a +2)(a -5) = 0

  a = -2 or +5 . . . . . . by the zero-product rule

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The intercept form of the equation for a line is ...

  x/p +y/q = 1 . . . . . . x-intercept = p; y-intercept = q

Possible lines through these intercept points are ...

  x/1 +a/-2 = 1   ⇒   2x -a -2 = 0

  x/-1 +a/-2 = 1   ⇒   2x +a +2 = 0

  x/1 +a/5 = 1   ⇒   5x +a -5 = 0

  x/-1 +a/5 = 1   ⇒   5x -a +5 = 0

These can be paired up two ways. The one of interest can be found by considering the resulting ax term.

  (5x +a -5)(2x +a +2)   ⇒   ax term is 7ax

  (5x -a +5)(2x -a -2)   ⇒   ax term is -7ax, as in the given expression

The factorization is (5x -a +5)(2x -a -2).