Factoring is the visible work and it is not where the marks go. The step that produces the answers is the sentence after it — a product is zero only when one of the things being multiplied is zero — so each bracket is set to zero and solved on its own.
Why this sheet works
The commonest error on this standard is reading (x − 3)(x + 5) = 0 and writing x = −3, 5: each bracket's number copied out with its sign attached. The sign turns over, and it only turns over when you actually solve the bracket. That is why one setting hands the brackets straight to the student — it isolates the zero-product step from the factoring, and a student can have the second and still get the first wrong.
How this one works
One question from this sheet, worked through a step at a time.
(x + 4)(x + 19) = 0
(x + 4)(x + 19) = 0
(x + 4)(x + 19) = 0
x + 4 = 0orx + 19 = 0
x + 4 = 0→x = −4
x + 19 = 0→x = −19
x = −4orx = −19
The left side is already a product of two brackets, and the right side is zero. That is the useful shape, so no factoring is needed here.
Now the step that gives the answers: a product is zero ONLY when one of the things being multiplied is zero. Two numbers that multiply to zero — one of them has to be zero. So either (x + 4) is zero, or (x + 19) is zero.
Solve each one. x + 4 = 0 gives x = −4, and x + 19 = 0 gives x = −19. Notice the sign turns over: a + in the bracket gives a NEGATIVE answer.
So x = −4 or x = −19. Either one makes the equation true, and it is "or" rather than "and" — x cannot be both at once.
